34. Frieden und Friedenssymboliken in der Bildenden Kunst

SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS: BEYOND THE FRACTIONAL LAPLACIAN IVAN BIOČIĆ∗ ZORAN VONDRAČEK∗ AND VANJA WAGNER∗ arXiv:2010.06240v3 [math.AP] 5 Feb 2021 Abstract. We study semilinear problems in general bounded open sets for non-local operators with exterior and boundary conditions. The operators are more general than the fractional Laplacian. We also give results in case of bounded C 1,1 open sets. AMS 2020 Mathematics Subject Classification: Primary 35R11; Secondary 31B25, 31C05, 35J61, 45K05, 60J35 Keywords and phrases: Semilinear differential equations, non-local operators 1. Introduction Let D ⊂ Rd , d ≥ 2, be a bounded open set, f : D × R → R a function, λ a signed measure on D c = Rd \ D and µ a signed measure on ∂D. In this paper we study the semilinear problem −Lu(x) = f (x, u(x)) u = λ WD u = µ in D in D c on ∂D. (1.1) The operator L is a second-order operator of the form L = −φ(−∆) where φ : (0, ∞) → (0, ∞) is a complete Bernstein function without drift satisfying certain weak scaling conditions. The operator L can be written as a principal value integral Z Lu(x) = P.V. (u(y) − u(x))j(|y − x|) dy, Rd where the singular kernel j is completely determined by the function φ. In case φ(t) = tα/2 , α ∈ (0, 2), −L is the fractional Laplacian (−∆)α/2 and the kernel j(|y − x|) is proportional to |y − x|−d−α . The operator WD is a boundary trace operator first introduced in [14] in the case of the fractional Laplacian, and extended to more general non-local operators in [10] – see Subsection 2.6 for the precise definition. Motivated by the recent preprint [3] we consider solutions of (1.1) in the weak dual sense, cf. Definition 3.1, and show that for bounded C 1,1 open sets this is equivalent to the notion of weak L1 solution as in [1, Definition 1.3]. For the nonlinearity f throughout the paper we assume the condition (F) f : D × R → R is continuous in the second variable and there exist a function ρ : D → [0, ∞) and a continuous function Λ : [0, ∞) → [0, ∞) such that |f (x, t)| ≤ ρ(x)Λ(|t|). Semilinear problems for the Laplacian have been studied for at least 40 years and we refer the reader to the monograph [42] for a detailed account. The study of semilinear problems for non-local operators is more recent and is mostly focused on the fractional Laplacian, see [24, 17, 1, 2, 5, 4, 6, 14, 23]. One of the important differences between the local and non-local equations is that in the non-local case the boundary blow-up solutions are possible even for ∗ Research supported in part by the Croatian Science Foundation under the project 4197. 1 2 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER linear equations. To be more precise, there exist non-negative harmonic functions for the operator L that blow up at the boundary. In this paper we will restrict ourselves to the so called moderate blow-up solutions, that is those bounded by harmonic functions with respect to the operator L. This restriction is a consequence of the problem (1.1) itself, namely of the boundary trace requirement on the solution. In this respect we follow [1, 14] where the boundary behavior of solutions was also imposed. Note that in [1] the theory was developed for the fractional Laplacian in a bounded C 1,1 open set D, while [14] extends part of the theory to regular open sets. This extension was possible mainly due to potential-theoretic results from [15]. The goal of this paper is to generalize results from [1, 14] and at the same time to provide a unified approach. The first main contribution of the paper is that we replace the fractional Laplacian with a more general non-local operator. This is possible due to potential-theoretic and analytic properties of such operators developed in the last ten years. For the most recent development see [8, 9, 28, 29, 30]. Here we single out the construction of the boundary trace operator for the operator L in the recent preprint [10]. The second main contribution is that we obtain some of the results from [1] (which deals with C 1,1 open sets) for regular open subsets of Rd . To achieve this goal we combine methods from [1] with those of [14]. Let us now describe the content of the paper in more detail. In the next section we introduce notions relevant to the paper and recall known results. This includes the notion of the nonlocal operator L, the underlying stochastic process X = (Xt )t≥0 and its killed version upon exiting an open set, the notion of harmonic function with respect to X (or L), and the Green, Poisson and Martin kernel of an arbitrary open subset of Rd . We explain accessible and inaccessible boundary points and its importance to the theory. The boundary trace operator is introduced in Subsection 2.6, cf. Definition 2.1. The section ends with several auxiliary results about continuity of Green potentials. Section 3 is central to the paper and contains two main results on the existence of a solution to the semilinear problem (1.1) in arbitrary bounded open sets. The first result, Theorem 3.6, can be thought of as a generalization of [1, Theorem 1.5]. It assumes the existence of a subsolution and a supersolution to the problem (1.1) and gives several sufficient conditions for the existence of a solution. As in almost all existence proofs of semilinear problems, the solution is obtained by using Schauder’s fixed point theorem. As a corollary of the third part of that theorem, in Corollary 3.8 we obtain a generalization of the main result of [14]. Theorem 3.10 deals with nonpositive function f and is a generalization of [1, Theorem 1.7]. The main novelty of our approach is contained in using Lemma 3.9 to approximate a nonnegative harmonic function by an increasing sequence of Green potentials. This replaces the approximation used in [1] which works only in smooth open sets. In the last two sections we look at the semilinear problem for L in bounded C 1,1 open sets and at some related questions. In Section 4 we first recall the notion of the renewal function whose importance comes form the fact that it gives exact decay rate of harmonic functions at the boundary. We then state known sharp two-sided estimates for the Green function, Poisson kernel, Martin kernel and the killing function in terms of the renewal function. Subsection 4.3 may be of independent interest - there we give the boundary behavior of the Green potential and the Poisson potential of a function of the distance to the boundary. We next provide a sufficient integral condition (in terms of the renewal function) for a function of the distance to the boundary to be in the Kato class. In Subsection 4.6 we invoke a powerful result from [28] to show the existence of generalized normal derivative at the boundary which is used in the equivalent formulation of the weak dual solution. We end the section with a discussion on the relationship of the boundary trace operator WD with the boundary operator used in [1, 2]. SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 3 The last section revisits Theorem 3.10 and Corollary 3.8 in bounded C 1,1 sets. In case when f (x, t) = W (δD (x))Λ(t) for some function W , we give a sufficient and necessary integral condition for (a version of) Theorem 3.10 to hold in terms of W , Λ and the renewal function. Building on Lemma 4.5 we next give a sufficient condition for Corollary 3.8 to hold in a bounded C 1,1 set. Finally, we end by establishing Theorem 5.3 that extends Corollary 3.8 for nonnegative nonlinearities f . This result generalizes [1, Theorem 1.9]. The Appendix has two parts. In the first part we provide a proof of Lemma 3.9 in a more general context. In the second part, we give quite technical proofs of Propositions 4.1 and 4.2. The proof of Proposition 4.1 is modeled after the proof of [3, Theorem 3.4], while the proof of Proposition 4.2 is somewhat simpler. We end this introduction with a few words about notation. Let D ⊂ Rd be an open set. Then Cb (D) denotes the family of all bounded continuous real valued functions on D, C0 (D) the family of all continuous functions vanishing at infinity (i.e. f ∈ C0 (D) if for every ǫ > 0 there exists a compact subset K ⊂ D such that |f (x)| < ǫ for all x ∈ D \ K), Cc∞ (D) the family of all infinitely differentiable functions with compact support, B(D) Borel measurable functions on D, and Bb (D) bounded Borel measurable functions on D. If µ is a measure on D, then L1 (D, µ) denotes integrable functions, L1loc (D, µ) locally integrable functions and L∞ (D, µ) essentially bounded functions on D. In case when µ is the Lebesgue measure on D, we simply write L1 (D), L1loc (D) and L∞ (D). Denote by ∂D the boundary c of D, δD (x) = dist(x, ∂D) if x ∈ D, and δDc (z) = dist(z, ∂D) if z ∈ D . For U ⊂ D open, U ⊂⊂ D denotes that the closure U is contained in D. For A ⊂ Rd , M(A) denotes σ-finite signed measures on A and |λ| denotes the variation of λ ∈ M(A). For two positive functions f and g, f  g means that the quotient f /g stays bounded from above by a positive constant, and f ≍ g that the quotient f /g stays bounded between two positive constants. Finally, unimportant constants in the paper will be denoted by small letters c, c1 , c2 , . . . , and their labeling starts anew in each new statement. More important constants we denote by a big letter C, where e.g. C(a, b) means that the constant C depends only on parameters a and b. 2. Preliminaries 2.1. The process and the jumping kernel. Let X = (Xt , Px ) be a pure jump Lévy process in Rd , d ≥ 2, with the characteristic exponent Ψ : Rd → C given by Z
Ψ(ξ) = 1 − eiξ·y + iξ · y1{|y|≤1} ν(dy), Rd R where ν is a measure on Rd \ {0} satisfying Rd (1 ∧ |y|2 )ν(dy) < ∞ – the Lévy measure . Thus the Fourier transform of the distribution of Xt is given by E0 eiξ·Xt = e−tΨ(ξ) , ξ ∈ Rd , t > 0. We further assume that Ψ(ξ) = φ(|ξ|2) where φ : [0, ∞) → [0, ∞) is a complete Bernstein function, cf. [45, Chapter 6]. This means that Z ∞ φ(λ) = (1 − e−λt )µ(t)dt, λ > 0, 0 R∞ where µ : (0, ∞) → (0, ∞) is a completely monotone function such that 0 (1 ∧ t)µ(t)dt < ∞. Thus, in fact, the process X is a subordinate Brownian motion with the Lévy measure ν(dx) = j(|x|)dx where j : (0, ∞) → (0, ∞) is given by Z ∞ 2 j(r) = (4πt)−d/2 e−r /(4t) µ(t)dt, r > 0. (2.1) 0 4 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER We will refer to the function j as the Lévy density, or the jumping kernel, or simply, the kernel. The function j is strictly positive, continuous, decreasing and satisfies limr→∞ j(r) = 0. The main example of the process satisfying the assumptions presented in this subsection is the isotropic α-stable process in Rd , α ∈ (0, 2). In this case Ψ(|ξ|) = |ξ|α, φ(λ) = λα/2 , and j(r) = C(d, α)r −d−α for some explicit constant C(d, α) > 0. The isotropic stable process enjoys the exact scaling property which in terms of the complete Bernstein function φ(λ) = λα/2 reads as φ(t)/φ(s) = (t/s)α/2 . A similar property is also needed for the subordinate Brownian motion X. Thus we introduce the following weak scaling hypothesis: (H): There exist R0 > 0, 0 < δ1 ≤ δ2 < 1 and constants a1 , a2 > 0 such that  δ1  δ2 t φ(t) t a1 ≤ a2 ≤ , t ≥ s ≥ R0 . s φ(s) s (2.2) The number R0 above is not important: If (H) holds with some R0 > 0, then it holds with any R > 0, but with different constants a1 , a2 (δ1 and δ2 of course remain the same). It is well known that under the assumption (H) the kernel j enjoys sharp two-sided estimates for small r > 0: For every R > 0 there exists C = C(R) ≥ 1 such that C −1 φ(r −2)r −d ≤ j(r) ≤ Cφ(r −2)r −d , 0 < r < R, (2.3) see for example [13, (15), Corollary 22].l Moreover, the following properties of j are known: there exists C = C(φ) > 0 such that j(r) ≤ Cj(r + 1), r > 1, (2.4) for every M > 0 there exists C = C(M, φ) > 0 such that j(r) ≤ Cj(2r), r ∈ (0, M), cf. [31, (2.11), (2.12)], and there exists C = C(φ) > 0 such that  n d j(r) ≤ Cj(r), r ≥ 1, n = 1, 2, dr (2.5) (2.6) cf. [12, Proposition 7.2]. Further, by [34, Lemma 4.3], for every r0 ∈ (0, 1), lim sup δ→0 r>r0 j(r) = 1. j(r + δ) (2.7) Properties (2.4)–(2.7) are used in some of the results that we quote later. 2.2. The semigroup, the operator and the potential kernel. For a bounded or nonnegative function u ∈ B(Rd ) and t ≥ 0, define Pt u(x) := Ex [u(Xt )]. Then (Pt )t≥0 is the semigroup corresponding to X. It is well known that this semigroup has the Feller property, i.e., Pt : C0 (Rd ) → C0 (Rd ). The space Cc∞ (Rd ) of infinitely differentiable functions with compact support is contained in the domain of the infinitesimal generator of the semigroup, and for u ∈ Cc∞ (Rd ) it holds that Z
Lu(x) = u(y) − u(x) − ∇u(x) · (y − x)1{|y−x|≤1} j(|y − x|) dy (2.8) Rd Z (u(y) − u(x)) j(|y − x|) dy. (2.9) = lim ǫ→0 |y−x|>ǫ In the familiar case of the isotropic stable process the operator L is the fractional Laplacian. SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 5 Under our assumption, the process X is also strongly Feller, i.e., Pt : Bb (Rd ) → Cb (Rd ). Indeed, by using (H), it easily follows that Z   E0 eiξ·Xt |ξ|n dξ < ∞ Rd for all n ≥ 1, cf. [38, (3.5)]. It follows from [43, Proposition 2.5(xii) and Proposition 28.1] that Xt has a density Z −d cos(x · ξ)e−tΨ(ξ) dξ, p(t, x) = (2π) Rd which is infinitely differentiable in x. This immediately implies the strong Feller property. For t > 0 and x, y ∈ Rd let p(t, x, y) := p(t, x − y).R Then p(t, x, y) are transition densities of X (or the heat kernel) in the sense that Pt f (x) = Rd p(t, x, y)f (y) dy. The process X is transient if it satisfies the Chung-Fuchs condition Z 1 d/2−1 λ dλ < ∞ . φ(λ) 0 This condition is satisfied for d ≥ 3 and we always impose it in case d = 2. Under transience one can define the potential kernel (or the Green function) by Z ∞ G(x) = p(t, x) dt < ∞. 0 Moreover, under the assumption (H), one has the following comparability for small x, cf. [33, Lemma 3.2(b)]: For every R > 0, there exists C = C(R) > 1 such that C −1 φ(|x|−2 )−1 |x|−d ≤ G(x) ≤ Cφ(|x|−2 )−1 |x|−d , |x| ≤ R. (2.10) 2.3. Harmonic functions. Let L1 = L1 (Rd , (1 ∧ j(|x|))dx). For an open set U ⊂ Rd , let τU = inf{t > 0 : Xt ∈ / U} be the first exit time from U. A function u ∈ L1 is said to be harmonic in an open set D ⊂ Rd if for every open U ⊂ U ⊂ D,   x ∈ U. u(x) = Ex u(XτU ) , The function u is regular harmonic in D if the above equality holds with D instead of U. If c u is harmonic in D and u = 0 in D , then u is said to be singular harmonic. We say that the scale invariant Harnack inequality is valid if there exists r0 > 0 and a constant c = c(r0 ) > 0 such that for every x0 ∈ Rd , every r ∈ (0, r0 ) and every function u : Rd → [0, ∞) which is harmonic in the ball B(x0 , r) it holds that u(x) ≤ cu(y) , x, y ∈ B(x0 , r/2). It is well known that the scale invariant Harnack inequality is valid under the weak scaling condition (H), cf. [25, Theorem 1, Theorem 7]. Moreover, nonnegative harmonic functions are locally Hölder continuous, [25, Theorem 2, Theorem 7]. Under condition (2.6) it is shown in [12, Theorem 4.9] that if u is harmonic in an open set D, then u ∈ C 2 (D). e by For u ∈ L1 define the distribution Lu Z e hLu, ϕi := u(x)Lϕ(x)dx , ϕ ∈ Cc∞ (Rd ). Rd e = 0 in D (as a distribution), Let D ⊂ Rd be open. Then u is harmonic in D if and only if Lu cf. [26, Lemma 3.1 and Lemma 3.3] and [10, Theorem 3.14 and Theorem 3.16]. 6 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER 2.4. Transition density and Green function for the killed process. Let D ⊂ Rd be an open set, and τD = inf{t > 0 : Xt ∈ / D}. The killed process X D (or part of the process X) is defined by XtD = Xt if t < τD , and XtD = ∂ if t ≥ τD . Here ∂ is an extra point called the cemetery. Every Borel function f on D is extended to ∂ by letting f (∂) = 0. The semigroup (PtD )t≥0 of the killed process X D is defined by PtD f (x) = Ex [f (XtD )] = Ex [f (Xt ), t < τD ], f ∈ Bb (D). For t > 0 and x, y ∈ D let pD (t, x, y) = p(t, x, y) − Ex [p(t − τD , XτD , y), τD < t]. It is well known that pD (t, ·, ·) is symmetric on D × D. By R the strong Markov property, pD (t, x, y) is the transition density of X D , i.e., PtD f (x) = D pD (t, x, y)f (y)dy. Moreover, by continuity of p(t, x, y), the Feller and the strong Feller property of X, one can show that pD (t, x, y) is jointly continuous, see [22, pp. 34-35] and [39, Lemma 2.2 and Proposition 2.3]. Continuity of pD (t, x, y) implies that the semigroup (PtD )t≥0 is strongly Feller. Let Z ∞ Z τD D GD f (x) = Pt f (x)dt = Ex f (Xt )dt . 0 0 be the potential operator of the killed process X D . This operator admits the symmetric density Z ∞ GD (x, y) = pD (t, x, y) dt , x, y ∈ D, 0 R which we call the Green function of X D . That is, GD f (x) = D GD (x, y)f (y) dy. We extend the definition of the Green function by GD (x, y) = 0 if x ∈ D c or y ∈ D c . By using Hunt’s switching identity, it is standard to derive that for every open U ⊂ D, GD (x, y) = Ex [GD (XτU , y)], x ∈ D, y ∈ D \ U. In particular, for a fixed y ∈ D \ U, the function x 7→ GD (x, y) is regular harmonic in U and harmonic in D \ {y}. Since harmonic functions are continuous, we get that x 7→ GD (x, y) is continuous in D \ {y}. By symmetry, y 7→ GD (x, y) is continuous in D \ {x}. If f : D → [0, ∞] such that GD f (x) < ∞, for some x ∈ D, it was shown in [10, Remark 2.4] that GD f < ∞ a.e. and GD f ∈ L1 ∩L1loc (D). In particular, when D is bounded GD f ∈ L1 (D). 2.5. Martin kernel and Poisson kernel. From now on we assume that D is a bounded open subset of Rd . The Poisson kernel of D with respect to the process X is defined by Z PD (x, z) := GD (x, y)j(|y − z|) dy x ∈ D, z ∈ D c . (2.11) D It is well known and follows from the Lévy system formula (see [34, (1.1)]) that PD (x, ·) is the density (with respect to the Lebesgue measure) of the exit distribution of X from D (restricted c to D ): Z Px (XτD ∈ A) = PD (x, z)dz , A c A⊂D . (2.12) c Furthermore, it was shown in [10, Proposition 3.1] that PD (x, z) is jointly continuous in D×D . It is well known that if D has a Lipschitz boundary, then Px (XτD ∈ ∂D) = 0, see [31, (5.5)], and thus the equality (2.12) holds for every A ⊂ D c . We say that z ∈ ∂D is accessible from D with respect to X if PD (x, z) = ∞ for some (equivalently, every) x ∈ D, and inaccessible otherwise. The notion of accessible boundary point was introduced in [15] in the context of the fractional Laplacian. In the very general setting, accessible points were studied in [37] and [40]. It is shown in [37, Subsection 4.1] that the subordinate Brownian motion X of our paper satisfies all the assumptions of [37], so we SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 7 are free to use results of that paper. We mention that in case of sufficiently regular boundary ∂D (Lipschitz boundary is fine), all boundary points are accessible. Let ∂M D ⊂ ∂D denote the set of all accessible boundary points of D. Fix x0 ∈ D. It is shown in [37, Lemma 3.4, Theorem 1.1] that for every accessible point z ∈ ∂D there exists MD (x, z) := GD (x, y) , y→z,y∈D GD (x0 , y) lim (2.13) and x 7→ MD (x, z) is harmonic with respect to X D (i.e. singular harmonic with respect to X). In fact, the above limit exists for all boundary points z, but x 7→ MD (x, z) is not harmonic in case of an inaccessible point z ∈ ∂D. The function MD (x, z) is called the Martin kernel of D with respect to X. It is shown in [10, Proposition 5.11] (cf. [15] for the case of the fractional Laplacian) that u : D → [0, ∞) is harmonic with respect to X D if and only if there exists a nonnegative finite measure µ on ∂M D such that Z u(x) = MD (x, z)µ(dz) . ∂M D R In that case µ is unique. We will use the notation MD µ(x) = ∂M D MD (x, z)µ(dz). Since MD µ is a singular harmonic function with respect to X, we have that MD µ ∈ C 2 (D), and also by [10, Remark 5.12], it is in L1 (D). We note further that MD µ ≡ ∞ in D if and only if µ is an infinite measure, see [10, Corollary 5.13]. For a nonnegative measure λ on D c , we define Z PD λ(x) := PD (x, z)λ(dz), x ∈ D. Dc If λ is a signed measure on D c such that PD |λ| < ∞ in D, then PD λ is defined by the same formula. Note that if PD |λ|(x) < ∞, for some x ∈ D, [10, Corollary 3.11 and Remark 3.6] yield that PD λ is finite and continuous on the whole D, and PD λ ∈ L1 (D). We can say something more about the measure that satisfies PD |λ| < ∞. Since PD (x, z) = ∞ for z ∈ ∂M D, PD |λ| < ∞ implies that the measure λ has no mass on ∂M D so λ can be c viewed as a measure on Rd \ (D ∪ ∂M D). Also, λ is finite on compact subsets of D since for c a compact K ⊂ D we have that K ∋ y 7→ PD (x, R y) is bounded and strictly positive. Let ν ∈ M(D) and set u(y) = GD ν(y) := D GD (y, v)ν(dv). Then for z ∈ D c we have u(z) = 0, hence Z Z Lu(z) = lim (u(y) − u(z))j(|y − z|) dy = GD ν(y)j(|y − z|) dy ǫ→0 |y−z|>ǫ Rd  Z Z GD (y, v)ν(dv) j(|y − z|) dy = D D  Z Z = GD (y, v)j(|y − z|) dy ν(dv) D D Z = PD (v, z) ν(dv), D if the last integral absolutely converges. In particular, if ν = δx for x ∈ D, where δx is the Dirac measure at x, then u(y) = GD (x, y) and LGD (x, ·)(z) = PD (x, z), 8 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER which gives an alternative expression for the Poisson kernel. Further, let ψ : D → R be bounded, u = GD ψ and λ ∈ M(D c ). Then  Z Z Z Lu(z) λ(dz) = PD (y, z)ψ(y) dy λ(dz) Dc Dc D Z  Z Z = ψ(y) PD (y, z) λ(dz) dy = ψ(y)PD λ(y) dy. (2.14) Dc D D 2.6. Boundary trace operator. Recall that x0 ∈ D is fixed. Let u : D → [−∞, ∞] and let U ⊂⊂ D be an open Lipschitz set such that x0 ∈ U. Let ηU u be a measure on Rd defined by Z  Z ηU u(A) = GU (x0 , z) j(|z − y|)u(y)dy dz, A ⊂ Rd . D\U A Following the approach in [14] we define the boundary trace operator WD . Definition 2.1. If the measures ηU |u| are bounded as U ↑ D and ηU u weakly converge to a measure µ as U ↑ D, then we denote µ by WD u, i.e. WD u := lim ηU u. U ↑D It was shown in [10, Lemma 5.2] that the measure WD u is concentrated on ∂D. Further, if µ is a finite signed measure on ∂M D, λ a σ-finite measure on D c such that PD |λ| < ∞, and f : D → R such that GD |f | < ∞, then by [10, Proposition 5.4 and Proposition 5.11], it holds that WD (MD µ) = µ, WD (PD λ) = WD (GD f ) = 0. (2.15) 2.7. Some auxiliary results about Green potentials. We keep assuming that D is a bounded open subset of Rd . Recall that a function q : D → [−∞, ∞] is said to be in the Kato class J with respect to X if the family of functions {GD (x, y)|q|(y) : x ∈ D} is uniformly integrable (with respect to the Lebesgue measure on D). Obviously, if |v| ≤ |q| and q ∈ J then v ∈ J . Next, we show that a function q : D → [−∞, ∞] satisfying Z lim sup |q(y)|φ(|x − y|−2)−1 |x − y|−d dy = 0 (2.16) ǫ→0 x∈Rd |x−y|<ǫ is in the Kato class J . Extend the function q to all of Rd by setting q(y) = 0 for y ∈ D c . Since GD (x, y) ≤ G(x, y), to show that q ∈ J it suffices to show that the family of functions {G(x, y)|q(y)| : x ∈ Rd } is uniformly integrable. By using (2.10), one can check that [46, (24), Lemma 5] holds true. Hence, we can apply [46, Theorem 1], which together with (2.10) implies that (2.16) is equivalent to Z t  lim sup Ex q(Xs ) ds = 0, (2.17) t↓0 x∈Rd 0 i.e. q is in the classical Kato class K(X) from [20] and [18]. By (2.16), q ∈ L1 (D) and therefore [18, Theorem 2.1(ii)] implies that q ∈ K∞ (X), i.e. Z d ∀ε > 0 ∃δ > 0 ∀B ∈ B(R ) such that λ(B) < δ ⇒ sup |q(y)|G(x, y)dy < ε. (2.18) x∈Rd B cf. [20, Definition 2.1(ii)]. Furthermore, by [20, Proposition 2.1], q ∈ K∞ (X) implies that q is Green bounded. Together with boundedness of D, [44, Theorem 16.8(iii)] gives that the family {G(x, y)|q(y)| : x ∈ Rd } is uniformly integrable, and therefore q ∈ J . Note that under (H), the condition (2.16) is satisfied for q ∈ Bb (Rd ), so every bounded function q is in the Kato class J . SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 9 Recall that the boundary point z ∈ ∂D is said to be regular (for D c ) if Pz (τD = 0) = 1. The set D is regular if every boundary point is regular. The same proof as in [14, Proposition 1.31] shows that if D is regular, then q ∈ J if and only if GD |q| ∈ C0 (D), and then GD q ∈ C0 (D). Let z ∈ ∂D be regular. Then for all x ∈ D, lim y→z,y∈D GD (x, y) = 0. A proof of this well-known result can be found in [36, Proposition 6.2]. The next result is also known – we include the proof for the sake of completeness. Lemma 2.2. Let D be a bounded open subset of Rd . Then GD 1 ∈ C(D) and limx→z GD 1(x) = 0 for every regular boundary point z ∈ ∂D. Proof. Let (xn )n∈N be any sequence of points in D. Since the constant function 1 is in J , the family {GD (xn , ·) : n ∈ N} is uniformly integrable. If xn → x ∈ D, then limn→∞ GD (xn , y) = GD (x, y) for a.e. y ∈ D, hence by Vitali’s theorem, see [44, Theorem 16.6 (ii) ⇐⇒ (iii)], it follows that Z Z lim GD (xn , y)dy = GD (x, y)dy , n→∞ D D proving that GD 1 ∈ C(D). If xn → z ∈ ∂D with z regular, R then limn→∞ GD (xn , y) = 0 for all y ∈ D. Again by Vitali’s theorem we get that limn→∞ D GD (xn , y) dy = 0. ✷ Denote by D reg the set of all regular boundary points of D. For δ > 0, let Dδ := {x ∈ D : dist(x, ∂D) > δ}. Lemma 2.3. Let v : D → [0, ∞) be a locally bounded function and ρ : D → [0, ∞) such that GD ρ ∈ C(D) and ρvGD 1 ∈ L1 (D). Then, for every x ∈ D it follows that Z |GD (x, y) − GD (w, y)|ρ(y)v(y)dy = 0. lim w→x D Proof. Let r > 0 such that B(x, r) ⊂ D and take a sequence (xn )n ⊂ B(x, r/2) such that xn → x. Since v is locally bounded in D, there exists a constant c1 > 0 such that v(y) ≤ c1 for all y ∈ B(x, r). Therefore, Z Z |GD (xn , y) − GD (x, y)|ρ(y)v(y)dy ≤ c1 |GD (xn , y) − GD (x, y)|ρ(y)dy D D Z + |GD (xn , y) − GD (x, y)|ρ(y)v(y)dy D∩B(x,r)c Since GD (xn , y)ρ(y) → GD (x, y)ρ(y) as n → ∞, for a.e. y ∈ D, by Vitali’s convergence theorem, [44, Theorem 16.6 (i) ⇐⇒ (iii)], it is enough to show that Z Z lim GD (xn , y)ρ(y)dy = GD (x, y)ρ(y)dy and n→∞ D D Z Z lim GD (xn , y)ρ(y)v(y)dy = GD (x, y)ρ(y)v(y)dy. n→∞ D∩B(x,r)c D∩B(x,r)c The first limit follows directly from the assumption GD ρ ∈ C(D). For the second integral, we will show that there exists a constant c2 > 0 such that GD (w, y) ≤ c2 GD 1(y) , w ∈ B(x, r/2), y ∈ D ∩ B(x, r)c . (2.19) Therefore, since ρvGD 1 ∈ L1 (D) and xn ∈ B(x, r/2), we can apply the dominated convergence theorem to obtain Z Z lim n→∞ GD (xn , y)ρ(y)v(y) dy = D∩B(x,r)c GD (x, y)ρ(y)v(y) dy. D∩B(x,r)c 10 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER It remains to show (2.19). First note that GD (·, y) are harmonic functions in B(x, r) for all y ∈ D ∩ B(x, r)c . By the Harnack principle, there exists c3 > 0 such that GD (w, y) ≤ c3 GD (x, y) , for all w ∈ B(x, r/2) and all y ∈ D ∩ B(x, r)c . (2.20) Let ψ : D → [0, 1] be a function with support in B(x, r/2). Then both GD ψ and GD (x, ·) are regular harmonic in D ∩ B(x, r)c and vanish in the sense of the limit on D reg and by definition c on D . Let z ∈ ∂D. By [34, Theorem 1.1], there exists a finite limit a(z) := lim y→z,y∈D GD (x, y) . GD ψ(y) Therefore, there exists a 0 < ǫ(z) < dist(B(x, r), ∂D)/2 such that GD (x, y) ≤ a(z) + 1 , GD ψ(y) for all y ∈ D ∩ B(z, ǫ(z)). By compactness of ∂D, there are finitely many points z1 , z2 , . . . , zn ∈ ∂D and δ > 0 such that ∂D ⊂ D \ Dδ ⊂ ∪nj=1 B(zj , ǫ(zj )). Thus for any y ∈ D \ Dδ it holds that GD (x, y) ≤ max (a(zj ) + 1) =: c4 . j=1,...,n GD ψ(y) (2.21) Further, since both GD ψ and GD (x, ·) are continuous (and strictly positive) on the compact set Dδ ∩ B(x, r)c , we get that GD (x, y) ≤ c5 , GD ψ(y) y ∈ Dδ ∩ B(x, r)c . Combining (2.20)–(2.22) together with GD ψ ≤ GD 1, we get (2.19). (2.22) ✷ Lemma 2.4. Let |g| ≤ f such that GD f ∈ C0 (D). Then GD g ∈ C0 (D). Proof. Let (xn )n ⊂ D be a sequence that converges to x ∈ D. We have Z |GD g(xn ) − GD g(x)| ≤ |GD (xn , y) − GD (x, y)||g(y)|dy D Z ≤ |GD (xn , y) − GD (x, y)|f (y)dy. (2.23) D Since GD (xn , y)f (y) → GD (x, y)f (y) as n → ∞ and GD f ∈ C0 (D) by Vitali’s theorem [44, Theorem 16.6 (i) ⇐⇒ (iii)] we have that the right-hand side of (2.23) tends to 0. Hence, GD g ∈ C(D). To see that GD g ∈ C0 (D) it is enough to notice that 0 ≤ |GD g(x)| ≤ GD f (x) in D so when x → z ∈ ∂D we have GD g(x) → 0. ✷ 3. The semilinear problem in bounded open set Let us now turn to the semilinear problem. For functions f : D × R → R and u : D → R let fu : D → R be a function defined by fu (x) = f (x, u(x)). SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 11 Definition 3.1. Let f : D × R → R be a function, λ ∈ M(D c ) and µ ∈ M(∂D) a measure concentrated on ∂M D, such that PD |λ| + MD |µ| < ∞ on D. A function u ∈ L1 (D) is called a weak dual solution to the semilinear problem −Lu(x) = f (x, u(x)) u = λ WD u = µ if u satisfies the equality Z u(x)ψ(x)dx = in D in D c on ∂D Z f (x, u(x))GD ψ(x)dx Z Z + PD (x, z)ψ(x) dx λ(dz) Dc D Z Z + MD (x, z)ψ(x) dx µ(dz), D (3.1) D ∂M D (3.2) D for every ψ ∈ Cc∞ (D). If in the equation above we have ≥ (≤) instead of the equality and the inequality holds for every nonnegative ψ ∈ Cc∞ (D), we say that u is a supersolution (subsolution) to (3.1). Remark 3.2. (i) Recall from Subsections 2.4 and 2.5 that if PD |λ|(x) + MD |µ|(x) < ∞ for some x ∈ D, then PD |λ|(x) + MD |µ|(x) < ∞ for all x ∈ D. Also, since PD |λ| < ∞, λ is a measure on Rd \ (D ∪ ∂M D), see Subsection 2.5, so conditions in (3.1) in D c and on ∂D are indeed complementary. (ii)Note that by Fubini’s theorem and symmetry of GD , the above definition implies that the weak dual solution u of (3.1) satisfies u(x) = GD fu (x) + PD λ(x) + MD µ(x), for almost every x ∈ D. Moreover, if we set g = PD λ + MD µ, then (3.2) is equivalent to Z Z Z u(x)ψ(x)dx = f (x, u(x))GD ψ(x)dx + g(x)ψ(x)dx. (3.3) D D D L1loc (D) Also, suppose that u ∈ satisfies (3.1). This also implies that u = GD fu + PD λ + MD µ a.e. in D. Since GD fu , PD λ, MD µ ∈ L1 (D), see Subsections 2.4 and 2.5, we have u ∈ L1 (D), i.e. every function that satisfies (3.1) must be in L1 (D). Before we show an existence and uniqueness theorem for a wide class of problems we show an auxiliary result. For a Borel set A ⊂ D and x ∈ A, let ωAx (dz) := Px (XτAR ∈ dz) denote the harmonic measure. If u : Rd → [−∞, ∞], let PA u(x) := Ex [u(XτA )] = Rd u(y)ωAx (dy) whenever the integral makes sense. We also recall that GA (x, y) = 0 if y ∈ / A. Finally, if the d function u is defined only on D, we extend it to all of R by setting u(x) = 0 for x ∈ / D, and denote the extended function by u1D . Lemma 3.3. Let D be an open bounded set in Rd , f : D → [−∞, ∞] a function on D and λ ∈ M(D c ) such that GD |f |(x0 ), PD |λ|(x0 ) < ∞ for some x0 ∈ D. Let u be a function on D satisfying u(x) = GD f (x) + PD λ(x) for a.e. x ∈ D and A ⊂ D an open set. Then for a.e. x ∈ A, u(x) = GA f (x) + PA (u1D )(x) + Z Dc PA (x, y)λ(dy). (3.4) 12 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER Proof. First recall that if GD |f |(x0 ), PD |λ|(x0 ) < ∞ for some x0 ∈ D then GD |f |(x), PD |λ|(x) < ∞ for almost every x ∈ D, see Subsections 2.4 and 2.5. By the strong Markov property we have that Z GD (x, y) = GA (x, y) + D\A and then (2.11) implies that PD (x, y) = PA (x, y) + Z D\A GD (z, y)ωAx (dz), PD (z, y)ωAx (dz), x ∈ A, y ∈ D, x ∈ A, y ∈ D c . Therefore, for a.e. x ∈ A we have Z Z Z u(x) = GD (x, y)f (y)dy + GD (x, y)f (y)dy + PD (x, y)λ(dy) A D\A Dc Z Z Z = GA (x, y)f (y)dy + GD (z, y)ωAx (dz)f (y)dy A A D\A Z Z Z x PD (x, y)λ(dy) + GD (z, y)ωA (dz)f (y)dy + D\A D\A Dc Z  Z Z = GA (x, y)f (y)dy + GD (z, y)f (y)dy ωAx (dz) A D\A D Z + PD (x, y)λ(dy) Dc Z Z = GA (x, y)f (y)dy + u(z)ωAx (dz) A D\A Z  Z Z x − PD (z, y)λ(dy) ωA (dz) + PD (x, y)λ(dy) D\A Dc Dc Z Z Z x = GA (x, y)f (y)dy + u(z)ωA (dz) + PA (x, y)λ(dy). A D\A Dc ✷ Remark 3.4. Let u = GD f + PD λ as above and set u = λ on D c . For an open set A ⊂ D with a Lipschitz boundary consider the linear problem −LuA = f in A, uA = u in Ac , and WA uA = 0 on ∂A. Then Lemma 3.3 says that uA = u in A. Proposition 3.5. Let D ⊂ Rd be a bounded open set and let f : D × R → R be a function which is nonincreasing in the second variable. Then the continuous weak dual solution to (3.1) is unique. Proof. Let u1 and u2 be two continuous solutions to (3.1). Remark 3.2(ii) yields that ui = GD fui + PD λ + MD µ a.e. on D, i = 1, 2, hence u1 − u2 = GD fu1 − GD fu2 a.e. on D. Note that A := {x ∈ D : u1 (x) > u2 (x)} is open and that f (x, u1 (x)) ≤ f (x, u2 (x)), x ∈ A, since f is nonincreasing. Using Lemma 3.3 we get for a.e. x ∈ A
0 < u1 (x) − u2 (x) = GA (fu1 − fu2 )(x) + PA (u1 − u2 )1D (x) ≤ 0 hence A = ∅. Similarly we get {x ∈ D : u2(x) > u1 (x)} = ∅. ✷ Let us recall the condition (F) on the function f : (F) f : D × R → R is continuous in the second variable and there exist a function ρ : D → [0, ∞) and a continuous function Λ : [0, ∞) → [0, ∞) such that |f (x, t)| ≤ ρ(x)Λ(|t|). SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 13 Theorem 3.6. Let D ⊂ Rd be a bounded open set and let f : D × R → R be a function satisfying the condition (F). Let λ ∈ M(D c ) such that PD |λ| < ∞ and µ ∈ M(∂D) be a finite measure concentrated on ∂M D. Assume that the nonlinear problem (3.1) admits a weak dual subsolution u ∈ L1 (D) ∩ C(D) and a weak dual supersolution u ∈ L1 (D) ∩ C(D) such that u ≤ u. Set g := PD λ + MD µ and h := |u| ∨ |u|. If one of the following conditions holds (i) µ ≡ 0, GD ρ ∈ C0 (D) and u, u ∈ L∞ (D) such that for every open subset A ⊂ D and a.e. x∈A u(x) ≤ GA fu (x) + PA (u1D )(x) + PA λ(x), (3.5) u(x) ≥ GA fu (x) + PA (u1D )(x) + PA λ(x); (3.6) (ii)µ ≡ 0, Λ is nondecreasing, GD (ρΛ(h)) ∈ C0 (D) and u and u satisfy (3.5) and (3.6), respectively; (iii)Λ is nondecreasing, GD (ρΛ(h)) ∈ C0 (D) and there exists a constant C > 0 such that, on D, GD (ρΛ(h)) ≤ C and u − g ≤ −C < C ≤ u − g; then (3.1) has a weak dual solution u ∈ L1 (D) ∩ C(D) satisfying u ≤ u ≤ u. (3.7) If, in addition, f is nonincreasing in the second variable, then u is a unique continuous weak dual solution to (3.1). Remark 3.7. Note that by Lemma 3.3 a supersolution u to the nonlinear problem (3.1) satisfies the condition (3.6) if, for example, u is a solution to the nonlinear problem −Lu(x) = f (x, u(x)) u = e λ WD u = 0 in D in D c on ∂D e < ∞ on D and λ ≤ λ e (for details see the proof of for some e λ ∈ M(D c ) such that PD |λ| Theorem 3.10 and the functions un,k ). Proof of Theorem 3.6. First note that by using (3.3) and (2.15), a function u ∈ L1 (D) is the solution to (3.1) if and only if u − g is the solution to the homogeneous problem −Lw(x) = f (x, w + g) w = 0 WD w = 0 in D in D c on ∂D. (3.8) Thus, we solve (3.8). For general v ∈ C0 (D), the function fv need not satisfy the Kato condition GD |fv | ∈ C0 (D), so we define a modification of f in the following way:   t > u(x) − g(x) f (x, u(x)), F (x, t) = f (x, t + g(x)), u(x) − g(x) ≤ t ≤ u(x) − g(x) (3.9)  f (x, u(x)), t < u(x) − g(x). Note that F is continuous in the second variable. Furthermore, if v ∈ C0 (D), then GD |Fv | ∈ C0 (D), (3.10) since • under (i), GD ρ ∈ C0 (D) and |F (x, v(x))| ≤ ρ(x) max Λ(y), y∈[0,M ] (3.11) where M := max{kuk∞ , kuk∞ } and c1 := maxy∈[0,M ] Λ(y) < ∞ so the claim now follows from Lemma 2.4; 14 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER • under (ii) and (iii), GD (ρΛ(h)) ∈ C0 (D) and |F (x, v(x))| ≤ ρ(x)Λ(|u(x)| ∨ |u(x)|) = ρ(x)Λ(h(x)), (3.12) and the claim again follows from Lemma 2.4. Next we consider an auxiliary problem −Lu(x) = F (x, u) u = 0 WD u = 0 in D in D c on ∂D, whose solution will be given by the Schauder fixed point theorem. To this end, • under (i), set C := kGD ρkL∞ (D) kΛkL∞ ([0,M ]) ; • under (ii), set C := kGD (ρΛ(h))kL∞ (D) ; • under (iii), let C be the constant from the assumption (iii); and let K = {v ∈ C0 (D) : kvk∞ ≤ C}. Define the operator T by Z T v(x) = F (y, v(y))GD (x, y)dy, v ∈ C0 (D). (3.13) (3.14) D From (3.10) we have T v ∈ C0 (D). We now prove the continuity of T . Suppose the opposite, i.e. suppose that there are ε > 0, (xn )n ⊂ D, (vn )n ⊂ C0 (D) and v ∈ C0 (D) such that ||vn − v||∞ → 0 and |T vn (xn ) − T v(xn )| ≥ ε, for all n ∈ N. Since D is compact there is x ∈ D and a subsequence of (xn )n denoted again by (xn )n such that xn → x. We have ε ≤ |T vn (xn ) − T v(xn )| ≤ |T vn (x) − T v(x)| + |T vn (xn ) − T vn (x)| + |T v(x) − T v(xn )|. (3.15) Note that if x ∈ ∂D, then T vn (x) = T v(x) = 0 by (3.14). Since F is continuous in the second variable using the dominated convergence theorem with bounds from (3.11) and (3.12) for the first term, for x ∈ D we have |T vn (x) − T v(x)| → 0 as n → ∞. For the second and the third term let us also look first at the case x ∈ ∂D. Note that from (3.11) and (3.12) we have • under (i) Z |T w(xn )| ≤ c1 GD (xn , y)ρ(y)dy = c1 GD ρ(xn ) → 0, as xn → x, w ∈ {v, vn }, D since GD ρ ∈ C0 (D); • under (ii) and (iii) Z |T w(xn )| ≤ GD (xn , y)ρ(y)Λ(h(y))dy = GD (ρΛ(h))(xn ) → 0, as xn → x, w ∈ {v, vn }, D since GD (ρΛ(h)) ∈ C0 (D). If x ∈ D then GD (xn , y) → GD (x, y) so using [44, Theorem 16.6 (i) ⇐⇒ (iii)] • under (i) Z |T w(xn ) − T w(x)| ≤ c1 |GD (xn , y) − GD (x, y)|ρ(y)dy → 0, as xn → x, w ∈ {v, vn }, D since GD ρ ∈ C0 (D); • under (ii) and (iii) Z |T w(xn ) − T w(x)| ≤ |GD (xn , y) − GD (x, y)|ρ(y)Λ(h(y))dy → 0, as xn → x, w ∈ {v, vn }, D since GD (ρΛ(h)) ∈ C0 (D). SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 15 Thus, we have a contradiction with (3.15), i.e. T is continuous. Also, from (3.11), (3.12) and the choice of constant C we get T (K) ⊂ K. We are left to prove that T (K) is a precompact subset of K. By Arzelà-Ascoli theorem it suffices to note that the functions {T v : v ∈ K} are equicontinuous by the same calculations as above. Hence by the Schauder fixed point theorem there is a function u ∈ K such that Z u(x) = F (y, u(y))GD (x, y)dy, D i.e. u is a weak dual solution to (3.13). It follows immediately from (3.9) that, if u − g ≤ u ≤ u − g, then u is also a weak dual solution to (3.8). Finally, we show that the obtained solution u to (3.13) is between u − g and u − g. In case of assumption (iii), this is obvious. Under (i) or (ii), set A = {x ∈ D : u(x) > u(x) − g(x)}. Note that Fu (y) = fu (y) for all y ∈ A and that A is an open subset of D, since both u and u − g are continuous on D. Then, for every x ∈ A, by (3.4) we have u(x) + g(x) = GA Fu (x) + PA ((u + g)1D )(x) + PA λ(x) ≤ GA fu (x) + PA (u1D )(x) + PA λ(x) ≤ u(x), where the first inequality comes only from the middle term and the second one is (3.6). This implies that A = ∅. By using (3.5), one can analogously show that {x ∈ D : u(x) ≤ u(x) − g(x)} = ∅. Uniqueness follows from Proposition 3.5. ✷ In the following corollary we extend the main result from [14] to our setting of more general non-local operators. Corollary 3.8. Let D ⊂ Rd be a bounded open set and let f : D × R → R be a function satisfying the condition (F) with Λ nondecreasing. Let λ ∈ M(D c ) such that PD |λ| < ∞ and µ ∈ M(∂D) a finite measure concentrated on ∂M D. Set g := PD λ + MD µ and g := PD |λ| + MD |µ|. Assume that GD ρ ∈ C0 (D), GD (ρΛ(2g)) ∈ C0 (D), and that either (a) Λ is sublinearly increasing, limt→∞ Λ(t)/t = 0, or (b) m is sufficiently small. Then the semilinear problem −Lu(x) = mf (x, u(x)) in D u = λ in D c (3.16) WD u = µ on ∂D has a weak dual solution u ∈ L1 (D) ∩ C(D) such that |u| ≤ g + C, for some C > 0. If, in addition, f is nonincreasing in the second variable, u is a unique continuous weak dual solution to (3.16). Proof. We use Theorem 3.6(iii) with mf instead of f and first choose the constant C > 0. Set r1 := supx∈D GD ρ(x) and r2 := supx∈D GD (ρΛ(2g))(x). By the assumption, we have that r1 < ∞ and r2 < ∞. If (b) holds, given any C > 0 we can find m small enough such that m(Λ(2C)r1 + r2 ) ≤ C. If (a) holds, then since Λ is sublinearly increasing, we can find C > 0 large enough so that again m(Λ(2C)r1 + r2 ) ≤ C. Let u := C +g, u := −u and h := |u|∨|u| = C +g. Clearly, u and u belong to L1 (D)∩C(D) and satisfy u − g ≤ −C < C ≤ u − g. We check that u is a supersolution of (3.16). Indeed, |GD (mfu ) + g| ≤ mGD |fC+g | + g ≤ mGD (ρΛ(C + g)) + g


≤ mGD ρ Λ(2C) + Λ(2g) + g ≤ m Λ(2C)r1 + r2 + g ≤ C + g = u. 16 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER In the same way we see that u is a subsolution. It remains to check that GD (mρΛ(h)) ∈ C0 (D) and GD (mρΛ(h)) ≤ C. By the same computations as above we have GD (mρΛ(h)) ≤ mΛ(2C)GD ρ + mGD ρΛ(2g)) (3.17) ≤ m(Λ(2C)r1 + r2 ) ≤ C. Since GD ρ ∈ C0 (D) and GD (ρΛ(2g)) ∈ C0 (D), by (3.17) and Lemma 2.4 we also have GD (mρΛ(h)) ∈ C0 (D). ✷ Uniqueness follows from Proposition 3.5. Our next goal is to extend Corollary 3.8 to a wider class of nonpositive functions f . First we show an additional auxiliary result. This result provides an approximation of a nonnegative harmonic function on D by an increasing sequence of potentials. It is a consequence of a rather well-known fact that we prove in the appendix, see Proposition 6.3. We can use this result because the semigroup (PtD )t≥0 is strongly Feller, the process X D is transient, nonnegative harmonic functions are excessive, and the potential GD 1 is continuous and satisfies 0 < GD 1 < ∞ on D. Lemma 3.9. Let h : D → [0, ∞) be a harmonic function with respect to the process X D . There exists a sequence (fek )k≥1 of nonnegative, bounded and continuous functions such that GD fek ↑ h. Theorem 3.10. Let D ⊂ Rd be a bounded open set. Let f : D × R → (−∞, 0] be a function that satisfies (F) with GD ρ ∈ C0 (D). Assume, additionally, that f (x, 0) = 0. Let λ ∈ M(Rd \ D) be a nonnegative measure such that PD λ < ∞ and µ ∈ M(∂D) be a finite nonnegative measure concentrated on ∂M D. Let g := PD λ + MD µ. If the semilinear problem (3.1) satisfies one of the following conditions: (i) µ ≡ 0; (ii)µ 6≡ 0, the function Λ is nondecreasing and ρΛ(g)GD 1 ∈ L1 (D); then the problem (3.1) has a nonnegative weak dual solution u ∈ L1 (D)∩C(D). If, in addition, f is nonincreasing in the second variable, then u is a unique continuous solution to (3.1). Proof. Let (fek )k be a sequence of nonnegative, bounded and continuous functions on D from Lemma 3.9 such that GD fek ↑ MD µ. Let (Kn )n be an increasing sequence of compact sets such c that Kn ↑ D . Then, for n ∈ N the measure λn (·) = λ(· ∩ Kn ) is a finite nonnegative measure c on D . Consider the following semilinear problem −Lu(x) = f (x, u(x)) + fek (x) u = λn WD u = 0 in D in D c on ∂D. (3.18) Since f (x, 0) = 0 and fek ≥ 0, u ≡ 0 is a subsolution to (3.18). Furthermore, since f is (n) nonpositive, as a supersolution to (3.18) we take the solution uk = GD fek + PD λn of the linear problem −Lu(x) = fek (x) in D u = λn in D c WD u = 0 on ∂D. (n) (n) Fix k ∈ N. Notice that uk ∈ C(D) and that, by Lemma 3.3, uk since λn is finite and sup x∈D,z∈Kn satisfies (3.4). Moreover, PD (x, z) ≤ j(dist(D, Kn )) sup GD 1(x) < ∞, x∈D SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 17 (n) uk is bounded. This means that we can apply Theorem 3.6(i) so that for n = 1 the problem (1) (3.18) has a solution u1,k ∈ C(D) ∩ L∞ (D) such that 0 ≤ u1,k ≤ uk . Note that since λ1 ≤ λ2 , u1,k is also a subsolution to the problem (3.18) for n = 2 such that (3.5) holds for every open subset A ⊂ D, that is for a.e. x ∈ A u1,k (x) = GA fu1,k (x) + GA fek (x) + PA (u1,k 1D )(x) + PA λ1 (x) ≤ GA fu (x) + GA fek (x) + PA (u1,k 1D )(x) + PA λ2 (x). 1,k (1) (2) Since u1,k ≤ uk ≤ uk , again by Theorem 3.6(i), there exists a solution u2,k ∈ C(D) ∩ L∞ (D) (2) to the problem (3.18) with λ2 on D c , such that u1,k ≤ u2,k ≤ uk . By iterating this procedure, we obtain an increasing sequence (un,k )n∈N of solutions to problems (3.18) for different n ∈ N. Moreover, the sequence (un,k )n∈N is dominated by the function u0k associated with the linear problem −Lu0k (x) = fek (x) in D 0 uk = λ in D c 0 WD uk = 0 on ∂D. Hence, the pointwise limit limn→∞ un,k = uk is well defined in D. We will now show that uk is a weak dual solution to the problem −Lu(x) = f (x, u(x)) + fek (x) u = λ WD u = 0 in D in D c on ∂D. (3.19) Take any ψ ∈ Cc∞ (D), ψ ≥ 0. Then by Fatou’s lemma and the continuity of the function f in the second variable, we get that Z Z − f (x, un,k (x))GD ψ(x)dx f (x, uk (x))GD ψ(x)dx ≤ − lim sup n→∞ D D Z Z un,k (x)ψ(x)dx + = − lim sup fek (x)GD ψ(x)dx D Z n→∞ D + PD λ(x)ψ(x)dx D Z Z Z e =− uk (x)ψ(x)dx + fk (x)GD ψ(x)dx + PD λ(x)ψ(x)dx, D D D where we used the monotone convergence theorem in the last line. The inequality above implies that uk is a weak dual subsolution to (3.19). To show that uk is also a supersolution of the same problem, set D ′ = supp ψ ⊂⊂ D and build a sequence (Dl )l∈N of sets with Lipschitz boundaries such that D ′ ⊂⊂ Dl ⊂⊂ D and Dl ↑ D. Obviously, ψ ∈ Cc∞ (Dl ), and both GDl ψ ↑ GD ψ and PDl λ ↑ PD λ pointwise in D. Also, notice that u0k = GD fek + PD λ is continuous, hence locally bounded. Furthermore, in Dl we have |f (x, un,k (x))|GDl ψ(x) ≤ Cρ(x)GDl ψ(x), R R where C := maxy∈Dl Λ(u0k (y)) < ∞, and ρGDl ψ ∈ L1 (D) since D ρGDl ψ = D ψGDl ρ ≤ R ψGD ρ < ∞. By using the dominated convergence theorem in the first equality and Lemma D 3.3 in the second, we have Z Z e [f (x, uk (x)) + fk (x)]GDl ψ(x)dx = lim [f (x, un,k (x)) + fek (x)]GDl ψ(x)dx n→∞ Dl Dl 18 IVAN BIOČIĆ  = lim  n→∞ Z ZORAN VONDRAČEK un,k (x)ψ(x)dx − Dl Z AND VANJA WAGNER PDl un,k (x)ψ(x)dx − Dl   Z Z ≤ lim  un,k (x)ψ(x)dx − PDl λn (x)ψ(x)dx Z Dl  PDl λn (x)ψ(x)dx n→∞ = Z Dl uk (x)ψ(x)dx + Dl Z Dl PDl λ(x)ψ(x)dx. Dl Letting l → ∞ we obtain Z Z Z e [f (x, uk (x)) + fk (x)]GD ψ(x)dx ≤ uk (x)ψ(x)dx + PD λ(x)ψ(x)dx, D D D which proves that uk is a supersolution, and therefore the solution to (3.19). Notice that for µ ≡ 0 we have fek ≡ 0 so we have found a solution to the problem (3.1) under the assumption (i). Suppose that we have a function Λ with properties as in the assumption (ii) of this theorem. With the Arzelà-Ascoli theorem we will now find a suitable subsequence of (uk )k that converges to a function u that is a solution to the problem (3.1). To this end first notice that uk is given by Z Z uk (x) = GD (x, y)[f (y, uk (y)) + fek (y)]dy + PD (x, y)λ(dy) D Dc Z = GD (x, y)f (y, uk(y))dy + GD fek (x) + PD λ(x). (3.20) D Since f is nonpositive, uk ≤ g = PD λ + MD µ so we have the pointwise boundedness of the family (uk )k . Since GD fek increases to the continuous function MD µ, by Dini’s theorem the convergence is locally uniform so the usual 3ε-argument gives equicontinuity of the family (GD fek )k at every point x ∈ D. Also, PD λ is continuous in D so it remains to analyse the first term. We have Z Z GD (x, y)f (y, uk(y))dy − GD (z, y)f (y, uk (y))dy D D Z |GD (x, y) − GD (z, y)|ρ(y)Λ(uk (y))dy ≤ D Z ≤ |GD (x, y) − GD (z, y)|ρ(y)Λ(g(y))dy. D Equicontinuity of the first term in (3.20) now follows from Lemma 2.3. Now by Arzelà-Ascoli theorem we extract a subsequence (ukl )l which converges pointwise to a continuous function u. Without loss of generality, assume that uk → u. It remains to prove that u is a weak solution of (3.1), i.e., for every ψ ∈ Cc∞ (D) Z Z Z Z u(x)ψ(x)dx = f (x, u(x))GD ψ(x)dx + PD λ(x)ψ(x)dx + MD µ(x)ψ(x)dx. (3.21) D D D D We know that uk satisfies Z Z Z Z uk (x)ψ(x)dx = f (x, uk (x))GD ψ(x)dx + PD λ(x)ψ(x)dx + GD fek (x)ψ(x)dx. D D D D (3.22) SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 19 Since uk → u pointwise and uk ≤ g, by the dominated convergence theorem the left-hand side of (3.22) converges to the left-hand side of (3.21). Furthermore, by the monotone convergence theorem the last term of (3.22) converges to the last term of (3.21). To show the convergence of the first term on the right-hand side, note that |f (x, uk (x))GD ψ(x)| ≤ c1 ρ(x)Λ(g(x))GD 1(x). Now the assumption (ii) implies boundedness in L1 (D), so the convergence follows from the dominated convergence theorem. Hence, u is a solution to the problem (3.1). Uniqueness follows from Proposition 3.5. ✷ Remark 3.11. (i) Note that the condition ρΛ(g)GD 1 ∈ L1 (D) from Theorem 3.10 is weaker than the condition GD (ρΛ(2g)) ∈ C0 (D) from Corollary 3.8. (ii)Recall that if D is regular then q ∈ J if and only if GD |q| ∈ C0 (D). Hence, if we assume that D is regular in Theorem 3.6 then we can equivalently assume ρ ∈ J and ρΛ(h) ∈ J instead of GD ρ ∈ C0 (D) and GD (ρΛ(h)) ∈ C0 (D), respectively. Obviously, a similar argument applies to Corollary 3.8 and Theorem 3.10. 4. Auxiliary results in bounded C 1,1 open sets 4.1. The renewal function. We start this section by introducing a function which plays a prominent role in studying the boundary behavior in C 1,1 open sets. Let Z = (Zt )t≥0 be a one-dimensional subordinate Brownian motion with the characteristic exponent φ(θ2 ), θ ∈ R. We can think of Z as one of the components of the process X. Let Mt := sup0≤s≤t Zs be the supremum process of Z and let L = (Lt )t≥0 be the local time of Mt − Zt at zero. We refer the readers to [7, Chapter VI] for details. The inverse local time L−1 := inf{s > 0 : Ls > t} is called the ascending ladder time process of Z. Define the t ascending ladder height process H = (Ht )t≥0 of Z by Ht := ML−1 = ZL−1 if L−1 < ∞ and t t t Ht = ∞ otherwise. The renewal function of the process H is defined as Z ∞ V (t) := P(Hs ≤ t) ds, t ∈ R. 0 Then V (t) = 0 for t < 0, V (0) = 0, V (∞) = ∞, and V is strictly increasing. The importance of the renewal function V lies in the fact that V|(0,∞) is harmonic with respect to the killed process Z (0,∞) . This fact was for the first time used in [32] in order to obtain the precise rate of decay of harmonic functions of d-dimensional subordinate Brownian motion. In the case of the isotropic α-stable process, it holds that V (t) = tα/2 . In general, the function V is not known explicitly, but under the weak scaling condition (H) it is known, see e.g. [32], that there is a constant C = C(R0 ) ≥ 1 such that C −1 φ(t−2 )−1/2 ≤ V (t) ≤ Cφ(t−2 )−1/2 , 0 < t < R0 . (4.1) For more general results, covering also the case R0 = ∞, see [41, Theorem 4.4 and Remark 4.7]. Note that (4.1) and weak scaling (2.2) of φ imply that for all R1 ≥ 1 there are constants 0<e a1 ≤ e a2 depending on R1 such that  δ1  δ2 t V (t) t e a1 ≤ ≤e a2 , 0 < s ≤ t ≤ R1 . (4.2) s V (s) s 20 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER 4.2. Estimates in C 1,1 open set. Recall that an open set D in Rd (d ≥ 2) is said to be a C 1,1 open set if there exist a localization radius R > 0 and a constant Λ > 0 such that for every z ∈ ∂D, there exist a C 1,1 function ψ = ψz : Rd−1 → R satisfying ψ(0) = 0, ∇ψ(0) = (0, . . . , 0), k∇ψk∞ ≤ Λ, |∇ψ(x)−∇ψ(z)| ≤ Λ|x−z|, and an orthonormal coordinate system CSz : y = (y1 , · · · , yd−1 , yd ) := (e y , yd ) with origin at z such that B(z, R) ∩ D = {y = (e y , yd ) ∈ B(0, R) in CSz : yd > ψ(e y )}. The pair (R, Λ) is called the characteristics of the C 1,1 open set D. We remark that in some literature, the C 1,1 open set defined above is called a uniform C 1,1 open set since (R, Λ) is universal for all z ∈ ∂D. From now until the end of this section let D be a bounded open C 1,1 set. It is well known that all boundary points of a C 1,1 open set are regular and accessible. Thus, ∂M D = ∂D. Recall that δD (x) denotes the distance of the point x ∈ D to the boundary ∂D, while δDc (z) c denotes the distance of z ∈ D to ∂D. Under the weak scaling condition (H) the following sharp two-sided estimates of the Green function, Martin kernel and the Poisson kernel are known. The comparability constant depends on the constants in (2.2) and the diameter of D. We give the estimates in terms of the renewal function V :    V (δD (x)) V (δD (y)) V (|x − y|)2 GD (x, y) ≍ 1∧ 1∧ , x, y ∈ D, (4.3) V (|x − y|) V (|x − y|) |x − y|d V (δD (x)) , x ∈ D, z ∈ ∂D, (4.4) MD (x, z) ≍ |x − z|d 1 V (δD (x)) c , x ∈ D, z ∈ D . (4.5) PD (x, z) ≍ d V (δDc (z))(1 + V (δDc (z))) |x − z| For (4.3) see [19, Theorem 7.3(iv)], (4.4) follows immediately from (2.13) and (4.3), while (4.5) is proved in [27, Theorem 1.3]. We will also need sharp two-sided estimates of the killing function Z j(|y − x|) dy, κD (x) := x ∈ D. (4.6) Dc It holds that κD (x) ≍ V (δD (x))−2 , x ∈ D. (4.7) The upper bound is straightforward and valid in any open set D, while the lower bound holds in open sets satisfying the outer cone condition, see e.g. [35, proof of Lemma 5.7]. 4.3. Green and Poisson potentials. In this subsection we state two results which should be of independent interest. The first one gives sharp two-sided estimates of the Green potential of the function x 7→ U(δD (x)) for a function U : (0, ∞) → [0, ∞) satisfying certain assumptions. The estimates are given in terms of the function U and the renewal function V . A similar result was shown in [3, Theorem 3.4]. Since our proof is modeled after and is very similar to the one in [3], we defer the proof to Appendix. The second result is a sort of a counterpart of the first one and gives sharp two sided estimates of the Poisson potential of the function e : (0, ∞) → [0, ∞). The proof of this second result is simpler e (δDc (z)) for a function U z 7→ U and will be also given in Appendix. To be more precise, let U : (0, ∞) → [0, ∞) be a function satisfying the following conditions: (U1) Integrability condition: It holds that Z 1 U(t)V (t) dt < ∞; (4.8) 0 SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 21 (U2) Almost nonincreasing condition: There exists C > 0 such that U(t) ≤ CU(s), 0 < s ≤ t ≤ 1; (4.9) (U3) Reverse doubling condition: There exists C > 0 such that U(t) ≤ CU(2t), t ∈ (0, 1); (4.10) (U4) Boundedness away from zero: U is bounded from above on [c, ∞) for each c > 0. We will refer to (U1)–(U4) as conditions (U). Note that if U(t) = t−β , β ∈ R, satisfies (4.8), then it satisfies (U). In particular, if the process X is isotropic α-stable, then (4.8) (hence (U)) is equivalent to −β + α/2 > −1. Proposition 4.1. Assume that a function U : (0, ∞) → [0, ∞) satisfies conditions (U). Then Z Z diam(D) U(t)V (t) V (δD (x)) δD (x) U(t)V (t) dt + V (δD (x)) dt . (4.11) GD (U(δD ))(x) ≍ δD (x) t 0 δD (x) Morover, if U is positive and bounded on every bounded subset of (0, ∞), then GD (U(δD ))(x) ≍ V (δD (x)). The asymptotic behavior of GD (U(δD )) is given by the largest term that appears in (4.11). In this generality, this is not easy to determine (but see [3, Theorem 3.4]). It will follow from the proof that GD (U(δD )) < ∞ if and only if (4.8) holds true. Clearly, if f : D → [0, ∞) is such that f (x) ≍ U(δD (x)), then GD f (x) is asymptotically equal to the right-hand side of (4.11). c Proposition 4.2. Let g : D → [0, ∞) be such that c e (δDc (y)), y ∈ D , g(y) ≍ U (4.12) e : (0, ∞) → [0, ∞). Assume that U e is bounded on every compact holds for some function U subset of (0, ∞) and satisfies Z ∞ e Z 1 e U (t) U (t) dt + dt < ∞ . (4.13) V (t)2 t 1 0 V (t) Then PD g(x) ≍ V (δD (x)) Z diam(D) 0 and PD g(x)  e (t) U dt, x ∈ D, V (t)(δD (x) + t) V (δD (x)) , x ∈ D. δD (x) (4.14) (4.15) e (t) = t−β , Remark 4.3. In the case of the fractional Laplacian and the power function U condition (4.13) becomes −α < β < 1 − α/2. Further, it is easy to see that for −β < α/2, the integral in (4.14) is comparable to δD (x)−β−α/2 , in the case β = −α/2 it is comparable to log(1/δD (x)), while for −β > α/2 it is comparable to a constant. We conclude that for g(y) = δDc (y)−β  −α < β < −α/2,  δD (x)−β , α/2 δD (x) log(1/δD (x)), β = −α/2, PD g(x) ≍  δD (x)α/2 , −α/2 < β < 1 − α/2. 22 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER 4.4. Boundary estimates of harmonic functions. Let σ denote the (d − 1)-dimensional Hausdorff measure on ∂D. It follows immediately from (4.4) and the estimate Z σ(dz) 1 ≍ , x ∈ D, d δD (x) ∂D |x − z| that Z V (δD (x)) , x ∈ D. δD (x) ∂D The following result appears as [11, Theorem 4.2] for the fractional Laplacian. MD σ(x) = MD (x, z)σ(dz) ≍ (4.16) Proposition 4.4. Let h ∈ L1 (∂D, σ) and let µ(dζ) = h(ζ)σ(dζ). If h is continuous at z ∈ ∂D, then MD µ(x) lim = h(z). (4.17) x→z,x∈D MD σ(x) Since the proof is essentially the same as the proof of [11, Theorem 4.2], we omit it. Proposition 4.4 has the following two consequences. Assume that h is nonnegative, continuous in D, not identically equal to zero, and set µ(dζ) = h(ζ)σ(dζ). Then since both MD µ and MD σ are continuous and D is bounded, we first conclude that there exists C = C(h) > 0 such that MD µ(x) ≤ CMD σ(x), x ∈ D. Secondly, there exist z ∈ ∂D, ǫ > 0, and C = C(h) > 0 such that MD µ(x) ≥ CMD σ(x), x ∈ D ∩ B(z, ǫ). Together with (4.16), these last two estimates imply that there is a constant C = C(h) > 1 such that V (δD (x)) , x ∈ D, (4.18) MD µ(x) ≤ C δD (x) V (δD (x)) MD µ(x) ≥ C −1 , x ∈ D ∩ B(z, ǫ). (4.19) δD (x) 4.5. Kato class revisited. In this subsection we give a sufficient condition for a function of the distance to the boundary to be in the Kato class J . First, note that by (2.11) and (4.6), we have that sup GD κD (x) ≤ 1. (4.20) x∈D Recall from (4.7) that κD (x) ≍ V (δD (x))−2 . The first part of the following result is an analogue of [14, Lemma 1.26]. Lemma 4.5. Let f : (0, ∞) → [0, ∞) be bounded on (0, M] for every M > 0, and limt→∞ f (t)/t = 0. (a) Let D be a bounded open set, h > 0 a locally bounded function on D such that h → ∞ at ∂D and Z GD (x, y)h(y) dy < ∞. sup x∈D (4.21) D Then f ◦ h ∈ J . (b) Let D be a bounded C 1,1 open set. Then x 7→ f (V (δD (x))−2 ) is in the Kato class J . (c) Let D be a bounded C 1,1 open set and let U : (0, ∞) → [0, ∞) satisfy condition (U4). If lim U(s)V (s)2 = 0, s→0 then x 7→ U(δD (x)) is in the Kato class J . (4.22) SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 23 Proof. (a) We will take R advantage of the equivalence of (i) and (ii) of [44, Theorem 16.8]. Denote c := supx∈D D GD (x, y)h(y) dy and let η > 0. There is t0 > 0 such that f (t)/t < ηc for every t ≥ t0 . Also, since h → ∞ at ∂D there is F ⊂⊂ D such that h > t0 on D \ F and since h is locally bounded we have M := supF h < ∞. Hence Z Z Z sup GD (x, y)f (h(y))dy ≤ sup GD (x, y)f (h(y))dy + sup GD (x, y)f (h(y))dy x∈D x∈D D x∈D F D\F ≤ (sup(0,M ] f ) sup Ex [τD ] + η < ∞, x∈D i.e. we have property (a) of (ii) in [44, Theorem 16.8]. Note that 1 ∈ J Rsince D is bounded so thereR is wη ∈ L1+ (D) and δ > 0 such that for all B ⊂ D with B wη < δ we have supx∈D B GD (x, y)dy < sup η f . Hence, for all such B it holds that (0,M ] Z Z Z sup GD (x, y)f (h(y))dy ≤ sup GD (x, y)f (h(y))dy + sup GD (x, y)f (h(y))dy x∈D x∈D B∩F B  ≤ (sup(0,M ] f ) sup x∈D x∈D B\F Z B  GD (x, y)dy  + η ≤ 2η. Since η was arbitrary we have (b) of (ii) in [44, Theorem 16.8.], i.e. f ◦ h ∈ J . (b) This follows immediately from (a) by using (4.20) and (4.7). (c) Define f (t) := U(V −1 (t−1/2 )) so that f (V (t)−2 ) = U(t). By the assumption on U, the function f is locally bounded. Moreover, by using the substitution t = V (s)−2 and the assumption (4.22), we get f (V (s)−2 ) f (t) = lim = lim U(s)V (s)2 = 0. lim s→0 V (s)−2 s→0 t→∞ t The claim now follows from (b). ✷ 4.6. Generalized normal derivative, modified Martin kernel and equivalent formulation of the weak dual solution. We now invoke the powerful recent result from [28] on boundary regularity of the solution of the equation −Lu(x) = ψ(x) u = 0 in D in D c where ψ is a bounded continuous function on D. It is proved in [28, Theorem 1.2] (see also [28, Theorem 3.10]), that u = GD ψ is the (viscosity) solution of the above equation, u/V (δD ) ∈ C γ (D), and u ≤ Ckψk∞ , V (δD ) C γ (D) for some constants γ > 0 and C > 0 depending only on d, D and φ. Here C γ (D) is the space of γ-Hölder continuous functions on D with the corresponding Hölder norm. It follows that u/V (δD ) can be continuously extended to D. In particular, for any bounded and continuous function ψ : D → R and for every z ∈ ∂D, there exists a finite limit d GD ψ(y) (GD ψ)(z) := lim . y→z,y∈D V (δD (y)) dV (4.23) We can think of d(GD ψ)/dV as the generalized normal derivative of the function GD ψ – instead of the distance function δD we use V (δD ). 24 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER If ψ is nonnegative and has compact support, then GD ψ is regular harmonic in D \ supp(ψ). By [34, Theorem 1.1], for any x ∈ D, there exists a finite limit GD ψ(y) . y→z,y∈D GD (x, y) lim Combining with (4.23), we see that for every x ∈ D and every z ∈ ∂D, there exists KD (x, z) := GD (x, y) . y→z,y∈D V (δD (y)) lim (4.24) We call KD (x, z) a modified Martin kernel, because given x0 ∈ D, we have that KD (x, z) = lim KD (x0 , z) y→z,y∈D GD (x,y) V (δD (y)) GD (x0 ,y) V (δD (y)) = lim y→z GD (x, y) = MD (x, z). GD (x0 , y) (4.25) Lemma 4.6. Let D be a bounded open set and let ψ : D → R be a bounded function with compact support and set u = GD ψ. Then Z d u(z) = KD (y, z)ψ(y) dy. dV D Proof. Let 2ǫ = dist(supp(ψ), ∂D), z ∈ ∂D, and x ∈ D such that |x − z| < ǫ. By using (4.3), we get that for y ∈ supp(ψ), V (|x − y|) V (diam(D)) GD (x, y) ≤c ≤c . d V (δD (x)) |x − y| ǫd Thus we can use the bounded convergence theorem to conclude from (4.24) that Z Z d GD ψ(x) GD (x, y) u(z) = lim = lim ψ(y) dy = KD (y, z)ψ(y) dy. x→z,x∈D V (δD (x)) x→z,x∈D D V (δD (x)) dV D ✷ Recall the weak dual formulation (3.2) of the semilinear problem (3.1). We will now rewrite the last two integrals in (3.2). Let ψ ∈ Cc∞ (D) and set ϕ = GD ψ. First, by using (2.14) we see that Z Z Z PD (x, z)ψ(x) dx λ(dz) = − (−Lϕ(z)) λ(dz). Dc D Dc Further, for µ ∈ M(∂D), let µ e(dz) := KD (x0 , z)µ(dz). By Lemma 4.6 and (4.25) Z Z Z Z Z d MD (x, z)ψ(x) dx µ e(dz) = KD (x, z)ψ(x) dx µ(dz) = ϕ(z)µ(dz) . ∂D D ∂D D ∂D dV Since ψ = −Lϕ, we see that the function u is a weak dual solution of the problem (3.1) if and only if Z Z Z Z d u(x)(−Lϕ)(x) dx = f (x, u(x))ϕ(x) dx − ϕ(z)µ(dz) . (−Lϕ(z)) λ(dz) + c D D D ∂D dV This formulation of a solution to the problem (3.1) in bounded C 1,1 open sets can be found in [1] in the case of the fractional Laplacian. SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 25 4.7. Another boundary operator. Following [1, Subsection 1.2] (see also [16, (2.2), Appendix B]) we R now introduce another boundary operator. For a measure µ ∈ M(∂D) set KD µ(x) := ∂D KD (x, z)µ(dz), x ∈ D. Note that by Remark 4.7(i), KD (x0 , ·) is continuous on ∂D. In the context of the Proposition 4.4, let µ(dζ) := f (ζ)σ(dζ), µ e(dζ) := KD (x0 , ζ)µ(dζ) and ν(dζ) := KD (x0 , ζ)σ(dζ). Then MD µ e(x) KD (x0 , z)f (z) KD µ(x) = lim = = f (z). x→z,x∈D MD ν(x) x→z,x∈D KD σ(x) KD (x0 , z) lim (4.26) For u : D → R and z ∈ ∂D, let ED u(z) := u(x) , x→z,x∈D KD σ(x) lim whenever the limit exists and is finite. Remark 4.7. We will need the following elementary calculations several times below. (i) Let u : D → R be a function and assume that for every z ∈ ∂D there exists a finite limit u e(z) := lim x→z,x∈D u(x). (4.27) Then, by applying the usual 2ε-argument, it follows that u e : ∂D → R is continuous. (ii)Assume further that D is bounded and u e(z) = 0 for all z ∈ ∂D. Then convergence in (4.27) is uniform in the sense that for every ǫ > 0 there exists a compact set F ⊂ D such that |u(x)| < ǫ for all x ∈ D \ F . Indeed, due to compactness of ∂D we easily find a finite cover V := ∪ni=1 B(zi , ri ), zi ∈ ∂D, of ∂D such that |u| ≤ ε on D ∩ V . Proposition 4.8. Let u : D → R. If ED u(z) exists for every z ∈ ∂D, then WD u exists and WD u(dz) = ED u(z)KD (x0 , z)σ(dz). Proof. Assume that ED u(z) exists for every z ∈ ∂D. By Remark 4.7(i), ED u is continuous on ∂D. Let ν(dz) = KD (x0 , z)σ(dz), µ(dz) = ED u(z)ν(dz) and Z Z v(x) := MD µ(x) = MD (x, z)ED u(z)ν(dz) = KD (x, z)ED u(z)σ(dz). ∂D ∂D By (4.26), for every z ∈ ∂D, v(x) = ED u(z), x→z,x∈D KD σ(x) lim hence ED v = ED u, so that limx→z,x∈D (u(x) − v(x))/KD σ(x) = 0 for every z ∈ ∂D. By Remark 4.7(ii), this implies that for every ǫ > 0 there exists a compact set F ⊂ D, such that |u(x) − v(x)| < ǫ, KD σ(x) for all x ∈ D \ F. Since KD σ is a nonnegative harmonic function, the same proof as [14, Lemma 1.16] gives that WD (u −v) = 0. Notice that the set of functions on D for which WD is defined is a vector space and WD is linear on that space. We conclude that WD u exists and WD u = WD v +WD (u−v) = WD v = WD (MD µ) = µ by (2.15). ✷ 26 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER 5. The semilinear problem in bounded C 1,1 open set 5.1. Corollary 3.8 revisited. Recall that in Corollary 3.8 we assumed that the function f : D × R → R satisfies (F) with Λ nondecreasing and that GD ρ ∈ C0 (D) and GD (ρΛ(2g)) ∈ C0 (D), where g = PD |λ| + MD |µ|. We give sufficient conditions for these assumptions in case of a bounded C 1,1 open set. We will additionally assume that ρ(x) = W (δD (x)) for a function W : (0, ∞) → [0, ∞) and that Λ satisfies the following doubling condition: There exists C ≥ 1 such that Λ(2t) ≤ CΛ(t), t > 0. (5.1) This implies that for all c1 > 1 there exists c2 = c2 (C, c1 ) such that Λ(c1 t) ≤ c2 Λ(t), t > 0, which can be rewritten as follows: For every e c1 ∈ (0, 1), there exists e c2 > 0 such that Secondly, assume that Λ(e c1 t) ≥ e c2 Λ(t), t > 0. (5.2) V (δD (x)) , x ∈ D. δD (x) By (4.15) and (4.18), this will be the case provided µ(dz) = h(z)σ(dz) for a continuous e is nonnegative, e (δDc (y)) where U function h : ∂D → R, and λ(dy) = g(y)dy with |g(y)|  U bounded on compact subsets of (0, ∞) and satisfies (4.13). Then we have   V (δD (x)) ρ(x)Λ(2g)(x) ≤ cW (δD (x))Λ , x ∈ D, δD (x) g(x)  for some c > 0. By using Lemma 4.5(c), we see that GD (ρΛ(2g)) ∈ C0 (D) if   V (t) V (t)2 = 0 , lim W (t)Λ t→0 t while GD ρ ∈ C0 (D) if limt→0 W (t)V (t)2 = 0. In the case of the fractional Laplacian, W (t) = t−β and Λ(t) = tp , these two conditions become β + p(1 − α/2) < α. 5.2. Theorem 3.10 in bounded C 1,1 open set. In this subsection we revisit Theorem 3.10(ii) in case of a bounded C 1,1 open set D. Recall that the assumptions of that theorem were that f : D × R → (−∞, 0] satisfies (F) with GD ρ ∈ C0 (D), f (x, 0) = 0 and the function Λ is nondecreasing. As in the previous subsection, we will additionally assume that ρ(x) = W (δD (x)) for a function W : (0, ∞) → [0, ∞) and that Λ satisfies the doubling condition (5.1). Proposition 5.1. Let D ⊂ Rd be a bounded C 1,1 open set. Let f : D × R → (−∞, 0] be a function that satisfies (F) with ρ(x) = W (δD (x)), where W : (0, ∞) → [0, ∞) is bounded away from zero, and such that Λ is a nondecreasing function satisfying the doubling condition (5.1). Assume that lim W (t)V (t)2 = 0 . (5.3) t→0 e : (0, ∞) → [0, ∞) is bounded on every compact subset of e (δDc (y))dy where U Let λ(dy) = U (0, ∞) and satisfies (4.13), and let µ(dz) = h(z)σ(dz) where h : ∂D → [0, ∞) is continuous and not identically equal to zero. If for some η > 0   Z η V (t) dt < ∞, (5.4) W (t)V (t)Λ t 0 then the semilinear problem (3.1) has a nonnegative weak dual solution u ∈ L1 (D) ∩ C(D). SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 27 Proof. We first note that the assumption (5.3) implies by Lemma 4.5(c) that ρ = W (δD ) ∈ J , and thus by Subsection 2.7, ρ ∈ C0 (D). Hence, in order to see that the semilinear problem (3.1) has a nonnegative solution it suffices to check that ρΛ(g)GD 1 ∈ L1 (D) where g = PD λ+MD µ. By (4.15) and (4.18) there exists a constant c1 > 0 such that g(x) ≤ c1 V (δD (x)) , δD (x) x ∈ D. Together with (5.1) this implies that     V (δD (x)) V (δD (x)) Λ(g(x)) ≤ Λ c1 ≤ c2 Λ , δD (x) δD (x) x ∈ D, for some c2 > 0. Therefore, there is c3 > 0 such that   V (δD (x)) ρ(x)Λ(g(x))GD 1(x) ≤ c3 W (δD (x))Λ V (δD (x)), x ∈ D. (5.5) δD (x)   By using boundedness of W (δD )Λ V δ(δDD ) V (δD ) inside D and the co-area formula near the boundary of D with the assumption (5.4) we see that   Z V (δD (x)) W (δD (x))Λ V (δD (x))dx < ∞. δD (x) D Now it follows from (5.5) that ρΛ(g)GD 1 ∈ L1 (D). ✷ Remark 5.2. (a) Proposition 5.1 allows a partial converse. Assume that f (x, t) = −W (δD (x))Λ(|t|) where Λ : (0, ∞) → (0, ∞) is a nondecreasing and unbounded function satisfying (5.1) Assume further that there exists η0 > 0 such that for all η ∈ (0, η0 ]   Z η V (t) dt = +∞. (5.6) W (t)V (t)Λ t 0 Let µ(dζ) = h(ζ)σ(dζ) with nonnegative continuous h, h 6= 0. Then the semilinear problem (3.1) does not have a nonnegative weak dual solution u ∈ L1 (D) such that ED u is well defined. To show this, suppose that there exists a nonnegative u that solves (3.1). Then u(x) = GD fu (x) + PD λ(x) + MD µ(x) a.e. Since by assumption, ED u exists, by Proposition 4.8 WD u also exists and WD u(dζ) = ED u(ζ)KD (x0 , ζ)σ(dζ). On the other hand, since u = GD fu + PD λ + MD µ, we have by (2.15) that WD u = WD (MD µ) = µ. Since µ(dζ) = h(ζ)σ(dζ), we get h(ζ) ED u(ζ) = σ(dζ) − a.e. KD (x0 , ζ) Choose z ∈ ∂D such that ED u(z) = h(z)/KD (x0 , z) > 0. Since ED u(z) = lim x→z,x∈D u(x) , KD σ(x) there exists ǫ > 0 such that 1 1 h(z) u(x) ≥ ED u(z)KD σ(x) = KD σ(x) = c1 KD σ(x), 2 2 KD (x0 , z) for all x ∈ D ∩ B(z, ǫ), where c1 = c1 (z, h) > 0. By using (4.3) and (4.24) to get the same estimate of KD (x, z) as the one of MD (x, z) in (4.4), we see in the same way as for (4.16) that KD σ(x) ≍ V (δD (x)) , δD (x) x ∈ D. 28 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER This implies that there exists c2 = c2 (z, h) > 0 such that u(x) ≥ c2 V (δD (x)) , δD (x) for all x ∈ D ∩ B(z, ǫ). Therefore, by using (5.2) this implies that for some c3 > 0     V (δD (y)) V (δD (y)) c3 Λ , for all y ∈ D ∩ B(z, ǫ). Λ(u(y)) ≥ Λ c2 δD (y) δD (y) Choose x ∈ D so that δD (x) ≍ |x − y| ≍ 1 whenever y ∈ D ∩ B(z, ǫ). By (4.3), there exists c4 > 0 such that GD (x, y) ≥ c4 V (δD (y)). Hence,   Z Z V (δD (y)) GD fu (x) = GD (x, y)f (y, u(y))dy ≤ −c3 c4 V (δD (y))W (δD (y))Λ dy. δD (y) D D∩B(z,ǫ) By use of the co-area formula it follows that the last integral is equal to some constant multiplied by   Z ǫ V (t) V (t)W (t)Λ dt. t 0 By (5.6) it follows that GD fu (x) = −∞ for points x in some open subset of D. This is a contradiction with GD fu > −∞ a.e. which follows from u ≥ 0, PD λ < ∞ and MD µ < ∞. (b) Note that the power function Λ(t) = tp is increasing and satisfies the doubling condition (5.1). Assume that W (t) = t−β and the underlying process is an isotropic α-stable process (so that V (t) = tα/2 ). Then (5.3) reads β < α, while the integral criterion (5.4) is equivalent to β + p(1 − α/2) < 1 + α/2. In case f (x, t) = −tp , we see that the problem (3.1) has a nonnegative solution u if p < (2 + α)/(2 − α), while in case p ≥ (2 + α)/(2 − α) a nonnegative solution u such that ED u is well defined does not exist. 5.3. Extending Corollary 3.8 to a wider class of nonnegative nonlinearities. Our next goal is to extend the results of Corollary 3.8 for nonnegative nonlinearities f . Unlike Theorem 3.10, this approach relies heavily on the estimates of Green and Poisson potentials in bounded C 1,1 domains. Theorem 5.3. Let f : D × R → [0, ∞) be a function, nondecreasing in the second variable, satisfying (F), with ρ = W (δD ) for some function W : (0, ∞) → [0, ∞), Λ nondecreasing and satisfying the doubling condition (5.1). Let λ ∈ M(D c ) be a nonnegative measure which e: e Dc ), where U is absolutely continuous with respect to the Lebesgue measure with density U(δ (0, ∞) → [0, ∞) is a function bounded on compact subsets of (0, ∞) satisfying (4.13). Let h : ∂D → [0, ∞) be a continuous function and let µ(dζ) = h(ζ)σ(dζ) be a measure on ∂D. Suppose that one of the following conditions hold:   (i) the function t 7→ W (t)Λ V t(t) , t > 0, satisfies the conditions (U); e satisfies the conditions (U). Morover assume that (ii) h ≡ 0 and the function W Λ(U) Z diam(D) e e (s) U(t) U dt  , (5.7) V (t)(s + t) V (s) 0 Z s e e (t))dt  sU(s) , W (t)V (t)Λ(U V (s) 0 (5.8) Z diam(D) e (t)) e (s) W (t)V (t)Λ(U U dt  , t V (s) s where the constants do not depend on 0 < s ≤ diam(D) . 2 SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 29 Then there exists a constant m1 > 0 such that for every m ∈ [0, m1 ] the semilinear problem −Lu(x) = mf (x, u(x)) u = λ WD u = µ in D c in D on ∂D (5.9) has a nonnegative weak dual solution u ∈ L1 (D). Proof. First we prove the theorem under assumption (i). Since f is nonnegative, the function u0 = PD λ + MD µ is a subsolution to (5.9). Recall from (4.15) and (4.18) that there exists a constant c1 > 0 such that V (δD (x)) u0 (x) ≤ c1 , x ∈ D. δD (x) Next we construct a supersolution u for (5.9) of the form u(x) = c2 V (δD (x)) , δD (x) i.e. find a constant c2 > c1 such that u(x) ≥ mGD fu (x) + u0 (x), x ∈ D, (5.10) for m small enough. To be exact, we show that for every c2 > c1 there exists m1 > 0 such that (5.10) holds for every m ∈ [0, m1 ]. Fix c2 > c1 . First note that by (F) and the doubling property (5.1) for Λ we have       V (δD (x)) V (δD (x)) V (δD (x)) f x, c2 ≤ W (δD (x))Λ c2 ≤ c3 W (δD (x))Λ δD (x) δD (x) δD (x) for some constant c3 > 0. Now by Proposition 4.1 there exists c4 > 0 such that    V (δD ) V (δD (x)) (x) ≤ c4 GD fu (x) ≤ c3 GD W (δD )Λ . δD δD (x) By choosing m1 = c2 −c1 c4 we get that for every m ≤ m1 mc4 + c1 V (δD (x)) = u(x) ≤ u(x). δD (x) c2 Now we can apply the classical iteration scheme in the following way: For k ∈ N let uk be the weak L1 solution to the linear problem mGD fu (x) + u0 (x) ≤ (mc4 + c1 ) −Luk (x) = mf (x, uk−1 (x)) uk = λ WD uk = µ in D c in D on ∂D. The constructed sequence (uk )k is nondecreasing and dominated by u. To see this, take x ∈ D. Since f is nonnegative, we have that u1 (x) − u0 (x) = mGD fu0 (x) ≥ 0. Furthermore, since f is nondecreasing in the second variable and u0 ≤ u, we have that u1 (x) = mGD fu0 (x) + u0 (x) ≤ mGD fu (x) + u0 (x) ≤ u(x). Assume now that uk−1 (x) ≤ uk (x) ≤ u(x) for some k ∈ N. This implies that fuk−1 (x) ≤ fuk (x) ≤ fu (x), so uk+1 (x) − uk (x) = mGD fuk (x) − mGD fuk−1 (x) ≥ 0 and uk+1(x) = mGD fuk (x) + u0 (x) ≤ mGD fu (x) + u0 (x) ≤ u(x). 30 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER The claim now follows by induction. Therefore, we can define a pointwise limit u := limk→∞ uk which, by the monotone convergence theorem and the continuity of f in the second variable, satisfies Z f (y, uk (y))GD (x, y)dy + u0 (x) u(x) = lim k→∞ D Z = lim f (y, uk (y))GD (x, y)dy + u0 (x) D k→∞ Z = f (y, u(y))GD (x, y)dy + u0 (x), D i.e. u is a weak L1 solution to (5.9). Next, consider the proof of the theorem under the assumptions (ii). Note that we only need to find a supersolution u ≥ u0 = PD λ satisfying (5.10). The rest of the proof then follows from the proof of (i). Note first that (4.14) and (5.7) imply that there exists a constant c5 > 0 such that e D (x)), x ∈ D. u0 (x) ≤ c5 U(δ Therefore, in this case we fix a constant c6 > c5 and show that the function u of the form e (δD (x)), u(x) = c6 U is indeed a supersolution to (5.9) for m small enough. As in the previous case, by (F) and the doubling property for Λ       e D (x)) ≤ W (δD (x))Λ c6 U e (δD (x)) ≤ c7 W (δD (x))Λ U e (δD (x)) f x, c6 U(δ for some constant c7 > 0. Now by Proposition 4.1 and (5.8) it follows that h  i e e (δD (x)). GD fu (x) ≤ c7 GD W (δD )Λ U (δD (x)) (x) ≤ c8 U By choosing m1 = c6 −c5 c8 we get that for every m ≤ m1 e (δD (x)) = mc8 + c5 u(x) ≤ u(x). mGD fu (x) + u0 (x) ≤ (mc8 + c5 )U c6 ✷ Assume that functions W and Λ satisfy (5.4), W satisfies conditions (U2)-(U4), and Λ is  non decreasing and satisfies the doubling condition (5.1). Then the function U(t) = W (t)Λ V t(t) satisfies conditions (U). Indeed, since W is almost nonincreasing and Λ is nondecreasing it follows that       Λ V t(t) (4.2) Λ e a2 V s(s) (5.1) W (t)Λ V t(t)       1, s < t ≤ 1.   W (s)Λ V s(s) Λ V s(s) Λ V s(s) Furthermore, since W satisfies the reverse doubling condition (4.10) and Λ is nondecreasing, we have that       W (t)Λ V t(t) Λ V t(t) Λ V t(t) (4.2) (5.1)         1, t ∈ (0, 1). (2t) (2t) W (2t)Λ V 2t Λ V 2t Λ e a1 2δ1 −1 V t(t) Finally, note that U is bounded away from zero, since both W and t 7→ V t(t) satisfy (U4) and Λ e satisfies conditions (U2)-(U4) is nondecreasing. Similarly, note that the function U = W Λ(U) e if we additionally assume that U satisfies (U2)-(U4). SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 31 −β1 Remark 5.4. (i) Consider the isotropic α-stable case and take Λ(t) = tp and  W (t)  = t for some p > 0 and β1 ≥ 0, as in Remark 5.2. The function U(t) = W (t)Λ V t(t) satisfies conditions (U) if and only if β1 + p(1 − α/2) < 1 + α/2. Hence, if f (x, t) = tp , then Theorem 5.3 holds for p < 2+α . 2−α e (t) = t−β2 , the function W Λ(U) e satisfies conditions (U) if and only if β1 + pβ2 < (ii)When U 1 + α/2. The condition (5.7) is satisfied for β2 < 1 − α/2. When β1 = 0 the conditions in (5.8) are satisfied when β2 (p − 1) ≤ α. Since β2 < 1 − α/2 we have that βα2 + 1 < 1+α/2 , so β2 α Theorem 5.3 states that the solution exists for p < β2 + 1. 6. Appendix 6.1. Approximation of excessive functions. Let (Xt , Px ) be a Hunt process on a locally compact space D and let (Pt )t≥0 denote its semigroup. Let U be the potential operator of X, that is Z Z Z ∞ Uf (x) = Ex ζ f (Xt ) dt = Ex 0 ∞ f (Xt )dt = 0 Pt f (x)dt. 0 Here ζ denotes the lifetime of the process. We assume that X is transient in the sense that there exists a nonnegative measurable function h such that 0 < Uh < ∞, see [21, p.86], and also that (Pt ) is strongly Feller. What follows essentially comes from [21, Section 3.2]. Recall that a measurable function f : E → [0, ∞] is said to be excessive relative to (Pt )t≥0 if f ≥ Pt f for all t ≥ 0 and f = limt→0 Pt f (see for example [21, Section 2.1]). Lemma 6.1. Suppose that f is excessive, Pt f < ∞ for all t ≥ 0 and limt→∞ Pt f = 0. Then there exists a sequence (gn )n≥1 of nonnegative measurable functions such that f =↑ lim Ugn . Moreover, if f is continuous and bounded, then one can choose gn to be continuous. Proof. This is proved as [21, Theorem 6, p.82]. The function gn is given by gn = n(f − P1/n f ). If f is bounded, then P1/n f is continuous (by the strong Feller property). If f is also continuous, then f − P1/n f is continuous. ✷ Remark 6.2. Transience is not needed in this result. The assumption Pt f < ∞ is satisfied if f < ∞ since Pt f ≤ f . The assumption limt→∞ Pt f = 0 is not satisfied for harmonic functions (since they are invariant). Proposition 6.3. Let f be excessive. If (Pt ) is transient, there exists a sequence (gn )n≥1 of bounded measurable functions such that f =↑ limn→∞ Ugn . Moreover, assume that there exists h > 0 such that 0 < Uh < ∞ and Uh is continuous. If f is continuous and (Pt ) is strongly Feller, then one can choose gn to be continuous. Proof. Let hn = nh with 0 < Uh < ∞. and put fn = f ∧ Uhn ∧ n. By [21, Theorem 8, p.104], fn is excessive (minimum of excessive function is excessive). Note that under additional assumptions, fn is continuous (and clearly bounded). By Lemma 6.1, there exists a sequence (gnk )k≥1 such that fn =↑ limk→∞ Ugnk . In fact, gnk = k(fn − P1/k fn ) ≤ kn . 32 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER Under additional assumptions, gnk are continuous. From the proof of Lemma 6.1, cf. [21, Theorem 6, p.82], Z 1/k Ugnk = k Ps fn ds ≤ n . 0 For each n, Ugnk increases with k (this is part of Lemma 6.1); for each k, Ugnk increases with n (this follows from fn ≤ fn+1 ). Now, by [21, Lemma 1, p.80], ↑ lim fn =↑ lim ↑ lim Ugnk =↑ lim Ugnn . n→∞ n→∞ n→∞ k→∞ On the other hand, by the same [21, Lemma 1, p.80] and monotone convergence ↑ lim fn =↑ lim ↑ lim Pt fn =↑ lim ↑ lim Pt fn ↑ lim Pt f = f. n→∞ n→∞ t↓0 t↓0 n→∞ t↓0 Therefore, by setting gn = gnn , f =↑ lim Ugn . n→∞ ✷ 6.2. Proofs of Propositions 4.1 and 4.2. Let ǫ > 0 be such that the map Φ : ∂D × (−ǫ, ǫ) → Rd defined by Φ(y, δ) = y + δn(y) defines a diffeomorphism to its image, cf. [3, Remark 3.1]. Here n denotes the unit interior normal. Without loss of generality assume that ǫ < diam(D)/20. Lemma 6.4. Let η < ǫ and assume that conditions (U) hold true. Then for any x ∈ D such that δD (x) < η/2, Z Z η
V (δD (x)) δD (x) GD U(δD )1(δD <η) (x) ≍ U(t)V (t) dt + V (δD (x)) U(t)V (t) dt δD (x) 0 0 Z η U(t)V (t) dt . (6.1) + V (δD (x)) t 3δD (x)/2
Further, GD U(δD )1(δD <η) (x) < ∞ if and only if the integrability condition (4.8) holds true. Proof. Let r0 := diam(D)/10. Fix x ∈ D as in the statement and define D1 D2 D3 D4 D5 = = = = = B(x, δD (x)/2) {y : δD (y) < η} \ B(x, r0 ) {y : δD (y) < δD (x)/2} ∩ B(x, r0 ) {y : 3δD (x)/2 < δD (y) < η} ∩ B(x, r0 ) {y : δD (x)/2 < δD (y) < 3δD (x)/2} ∩ (B(x, r0 ) \ B(x, δD (x)/2)). Thus we have that
GD U(δD )1(δD <η) (x) = 5 Z X j=1 GD (x, y)U(δD (y)) dy =: Dj 5 X Ij . j=1 Estimate of I1 : Under the almost nonincreasing condition (4.9) and the doubling condition (4.10) it holds that Z V (δD (x)) δD (x) 2 U(t)V (t) dt . (6.2) I1  U(δD (x))V (δD (x))  δD (x) 0 Indeed, let y ∈ D1 . Then δD (y) > δD (x)/2 > |y − x| implying that    V (δD (y)) V (δD (x)) 1∧ ≍ 1. 1∧ V (|x − y|) V (|x − y|) SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 33 Further, by using first (4.9) and then (4.10) we have that U(δD (y)) ≤ c1 U(δD (x)/2) ≤ c2 U(δD (x)). (6.3) Therefore, by using weak scaling of φ in the penultimate asymptotic equality, Z Z 1 V (|x − y|)2 U(δD (y)) I1 ≍ dy  U(δD (x)) dy d d −2 |x − y| D1 |y−x|<δD (x)/2 |x − y| φ(|x − y| ) Z δD (x) 1 1 dr ≍ U(δD (x)) ≍ U(δD (x))V (δD (x))2 .  U(δD (x)) −2 −2 ) rφ(r ) φ(δ (x) D 0 Finally, by (4.9) and the upper weak scaling (4.2) of V , Z δD (x) Z 1 U(δD (x))V (δD (x)) δD (x) V (t) dt U(t)V (t) dt  δD (x) 0 δD (x) V (δD (x)) 0 δ2  Z U(δD (x))V (δD (x)) δD (x) t dt  δD (x) δD (x) 0 ≍ U(δD (x))V (δD (x)). Estimate of I2 : Next, we show that I2 ≍ V (δD (x)) Z η U(t)V (t) dt . (6.4) 0 Let y ∈ D2 . Then r0 < |y − x| < diam(D) so that |y − x| ≍ 1. This implies that GD (x, y) ≍ V (δD (x))V (δD (y)). Therefore Z Z U(δD (y))V (δD (y)) dy ≍ V (δD (x)) U(δD (y))V (δD (y)) dy. I2 ≍ V (δD (x)) δD (y)<η D2 Finally, (6.4) follows by the co-area formula. In estimates for I3 , I4 and I5 we will use the change of variables formula based on a diffeophormism Φ : B(x, r0 ) → B(0, r0 ) satisfying Φ(D ∩ B(x, r0 )) = B(0, r0 ) ∩ {z ∈ Rd : z · ed > 0} Φ(y) · ed = δD (y) for any y ∈ B(x, r0 ), Φ(x) = δD (x)ed , see [3, page 38]. For the point z ∈ Rd+ = {z ∈ Rd : z · ed > 0} we will write z = (e z , zd ). Several times we also use the following integral: Z a sd−2 (1 + b/a)1−d ds = , a, b > 0. (6.5) d b(d − 1) 0 (b + s) Estimate of I3 : It holds that Z V (δD (x)) δD (x) U(t)V (t) dt . (6.6) I3 ≍ δD (x) 0 To see this, take y ∈ D3 . Then δD (y) ≤ δD (x)/2 implying |x − y| ≥ δD (x)/2, and thus GD (x, y) ≍ V (δD (x)) V (δD (y)) V (|x − y|)2 V (δD (x))V (δD (y)) = . d V (|x − y|) V (|x − y|) |x − y| |x − y|d Therefore I3 ≍ V (δD (x)) Z D3 ≍ V (δD (x)) Z U(δD (y))V (δD (y)) dy |x − y|d {0<zd <δD (x)/2}∩B(0,r0 ) U(zd )V (zd ) dz (|δD (x) − zd | + |e z |)d (6.7) 34 IVAN BIOČIĆ ≍ V (δD (x)) ZORAN VONDRAČEK Z |e z |<r0 ≍ V (δD (x)) Z r0 Z d−2 t 0 = V (δD (x)) Z 0 δD (x)/2 Z 0 s d−2 0 ≍ V (δD (x)) Z r0 /δD (x) Z U(zd )V (zd ) dzd dt (|δD (x) − zd | + t)d 1/2 0 sd−2 (1 + s)d 0 VANJA WAGNER U(zd )V (zd ) dzd de z (|δD (x) − zd | + |e z |)d δD (x)/2 r0 /δD (x) AND 1−d U(δD (x)h)V (δD (x)h) dh ds
d (1 − h) + s Z 1/2 ds U(δD (x)h)V (δD (x)h) dh Z 0 1/2 (1 + δD (x)/r0 ) U(δD (x)h)V (δD (x)h) dh d−1 0 Z 1/2 ≍ V (δD (x)) U(δD (x)h)V (δD (x)h) dh = V (δD (x)) 0 V (δD (x)) = δD (x) Z δD (x)/2 U(t)V (t) dt. 0 This proves the upper bound in (6.6). For the lower bound, note that by the upper weak scaling (4.2) of V and the almost nonincreasing condition (4.9), we have Z δD (x)/2 U(t)V (t) dt = 2 0 Z δD (x) U(t/2)V (t/2) dt ≥ 2 0 = c4 Z 0 Z δD (x) δD (x) −δ1 c3 U(t)e a−1 V (t) dt 1 2 U(t)V (t) dt . 0 Estimate of I4 : By applying the same change of variables as in I3 , we show that Z η U(t)V (t) I4 ≍ V (δD (x)) dt . t 3δD (x)/2 (6.8) Let y ∈ D4 . Then |x − y| ≥ δD (x)/2 and |x − y| ≥ δD (y)/3, hence GD (x, y) is of the form (6.7). By following the first five lines in the computation of I3 , we arrive at I4 ≍ V (δD (x)) ≍ V (δD (x)) = V (δD (x)) ≍ V (δD (x)) ≍ V (δD (x)) = V (δD (x)) Z Z Z Z Z Z r0 /δD (x) 0 η/δD (x) 3/2 η/δD (x) 3/2 η/δD (x) 3/2 η/δD (x) 3/2 η 3δD (x)/2 Z η/δD (x) U(δD (x)h)V (δD (x)h) dh ds
d 3/2 (h − 1) + s r0 Z U(δD (x)h)V (δD (x)h) (h−1)δD (x) r d−2 dr dh h−1 (1 + r)d 0 s d−2 U(δD (x)h)V (δD (x)h) (1 + (h − 1)δD (x)/r0 )1−d dh h−1 d−1 U(δD (x)h)V (δD (x)h) dh h−1 U(δD (x)h)V (δD (x)h) dh h U(t)V (t) dt . t SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 35 Estimate of I5 : Under the almost nonincreasing condition (4.9) and the doubling condition (4.10) it holds that Z V (δD (x)) δD (x) 2 U(t)V (t) dt . (6.9) I5  U(δD (x))V (δD (x))  δD (x) 0 Indeed, let y ∈ D5 . Then |x − y| > δD (x)/2 > δD (y)/3, hence GD (x, y) is of the form (6.7). Also, the estimate (6.3) and the analogous one with V hold true. Therefore Z U(δD (y))V (δD (y)) dy I5 ≍ V (δD (x)) |x − y|d D5 Z 1 2  U(δD (x))V (δD (x)) dy . d D5 |x − y| It is shown in [3, page 42] that the last integral is comparable to 1. This proves the first approximate inequality in (6.9), while the second was already proved in the estimate of I1 . The proof is finished by noting that I1 + I5  I3 . ✷ Lemma 6.5. Let η < ǫ and assume that conditions (U) hold true. There exists c(η) > 0 such that for any x ∈ D satisfying δD (x) ≥ η/2,
GD U(δD )1(δD <η) (x) ≤ c(η) . (6.10) Proof. Fix x ∈ D as in the statement and define D1 = {y : δD (y) < η/4}, D2 = {y : η/4 ≤ δD (y) < η}. Then
GD U(δD )1(δD <η) (x) = Estimate of J1 : We show that 2 Z X j=1 1 J1  η Z GD (x, y)U(δD (y)) dy =: Dj 2 X Jj . j=1 η U(t)V (t) dt. (6.11) 0 Let y ∈ D1 . Then δD (y) < η/4 ≤ δD (x)/2, hence by using |x−y| ≥ δD (x)−δD (y) we have that |x − y| > δD (y) and |x − y| > δD (x)/2. This implies that GD (x, y) satisfies (6.7). Therefore, Z U(δD (y))V (δD (y)) dy. J1 ≍ V (δD (x)) |x − y|d D1 By using the co-area formula we get (below dy denotes the Hausdorff measure on {δD (y) = t})  Z Z η/4 1 J1 ≍ dy dt. U(t)V (t) d 0 δD (y)=t |x − y| The inner integral is estimated as follows: For δD (y) = t it holds that |x − y| ≥ δD (x) − t, hence |x − y|−d ≤ (δD (x) − t)−d . The Hausdorff measure of {δD (y) = t} is larger than or equal to the Hausdorff measure of the sphere around x of radius δD (x) − t which is comparable to (δD (x) − t)d−1 . This implies that the inner integral is estimated from above by a constant times (δD (x) − t)−1 . Thus Z η/4 J1  U(t)V (t)(δD (x) − t)−1 dt. 0 36 IVAN BIOČIĆ ZORAN VONDRAČEK AND VANJA WAGNER If t < η/4, then t < δD (x)/2, implying δD (x)/2 < δD (x) − t < δD (x). Therefore, Z η/4 Z 1 2 η J1  U(t)V (t) dt  U(t)V (t) dt . δD (x) 0 η 0 Estimate of J2 : It holds that J2  U(η/4). (6.12) Let y ∈ D2 . By the almost nonincreasing condition (4.9) we have U(δD (y)) ≤ c1 U(η/4), hence Z Z V (|x − y|)2 V (|x − y|)2 dy  U(η/4) dy J2  U(δD (y)) |x − y|d |x − y|d η/4<δD (y)<η η/4<δD (y)<η Z V (|x − y|)2 ≤ U(η/4) dy  U(η/4). |x − y|d B(x,2diam(D)) The last estimate uses the fact that the integral is not singular. By putting together estimates for J1 and J2 , we see that there exists c2 > 0 such that   Z η
1 U(t)V (t) dt + U(η/4) =: c(η). GD U(δD )1(δD <η) (x) ≤ c2 η 0 ✷ Proof of Proposition 4.1: First we prove the statement under conditions (U). Fix some η < ǫ and treat it as a constant. Note that on {δD (y) ≥ η} it holds that U is bounded (by the assumption (U4)). Therefore GD (U(δD )1(δD ≥η) )(x) ≍ GD 1(x) ≍ V (δD (x)) .
By Lemma 6.5, if δD (x) ≥ η/2, then GD U(δD )1(δD <η) (x) ≤ c(η). Hence, GD (U(δD ))(x) ≍ 1, (6.13) δD (x) ≥ η/2. Since for δD (x) ≥ η/2 the right-hand side in (4.11) is also comparable to 1, this proves the claim for this case. Assume now that δD (x) < η/2. By Lemma 6.4 and (6.13) we have that GD (U(δD ))(x) = GD (U(δD )1(δD <η) )(x) + GD (U(δD )1(δD ≥η) )(x) Z Z η V (δD (x)) δD (x) U(t)V (t) dt + V (δD (x)) U(t)V (t) dt ≍ δD (x) 0 0 Z η U(t)V (t) + V (δD (x)) dt + V (δD (x)) t 3δD (x)/2 Z Z η U(t)V (t) V (δD (x)) δD (x) U(t)V (t) dt + V (δD (x)) dt . ≍ δD (x) t 0 δD (x) Clearly, in the last integral we can replace η by diam(D). Lastly, assume that the function U is bounded on every bounded subset of (0, ∞). Obviously, by (6.13), GD (U(δD ))(x)  GD 1(x) ≍ V (δD (x)). On the other hand, analogously as in (6.4), Z Z U(δD (y))GD (x, y) dy ≍ V (δD (x)) GD (U(δD ))(x) ≥ D2 η U(t)V (t) dt . 0 ✷ SEMILINEAR EQUATIONS FOR NON-LOCAL OPERATORS 37 Proof of Proposition 4.2: Fix η < ǫ. Let x ∈ D and r0 > δD (x) + η. We split D c into three parts, D1 = {z ∈ D c : δDc (z) ≥ η} D2 = {z ∈ D c ∩ B(x, r0 ) : δDc (z) < η} D3 = {z ∈ D c \ B(x, r0 ) : δDc (z) < η} and apply (4.12) to get that Z Z e PD g(x) ≍ U (δDc (z))PD (x, z)dz + D1 D2 =: I1 + I2 + I3 . e (δDc (z))PD (x, z)dz + U Z D3 e (δDc (z))PD (x, z)dz U Estimate of I1 : For z ∈ D c such that δDc (z) ≥ η, the estimate (4.5) is equivalent to PD (x, z) ≍ V (δD (x)) . V (δDc (z))2 δDc (z)d By applying this estimate and the co-area formula to I1 , we arrive to Z e (δDc (z)) U I1 ≍ V (δD (x)) dz d 2 D1 V (δD c (z)) δD c (z) Z ∞ e Z U(t) 1δ (w)=t dwdt ≍ V (δD (x)) V (t)2 td Dc D η Z ∞ e U (t) ≍ V (δD (x)) dt. V (t)2 t η As before, dw in the first two lines denotes the Hausdorff measure on δD (w) = t and we used that |{w ∈ D c : δDc (w) = t}| ≍ td−1 , t ≥ η. Estimate of I2 : First note that for z ∈ D c ∩ B(x, r0 ) estimate (4.5) implies that PD (x, z) ≍ V (δD (x)) . V (δDc (z))|x − z|d Next, as in the proof of Lemma 6.4 we will use the change of variables formula based on a diffeophormism Φ : B(x, r0 ) → B(0, r0) satisfying c Φ(D ∩ B(x, r0 )) = B(0, r0 ) ∩ {w ∈ Rd : w · ed < 0} c |Φ(z) · ed | = δD (z) for any z ∈ D ∩ B(x, r0 ), Φ(x) = δD (x)ed . Similarly as before, for the point w ∈ Rd− = {w ∈ Rd : w · ed < 0} we will write w = (w, e wd ). Therefore, by the change of variables given by the diffeomorphism Φ it follows that Z e Dc (z)) U(δ I2 ≍ V (δD (x)) dz d D2 V (δD c (z))|x − z| Z e U(−w d) ≍ V (δD (x)) dw. e d {w∈B(0,r0 ):−η<wd <0} V (−wd )(|δD (x) − wd | + |w|) Next, we apply the substitution wd = −t and switch to polar coordinates for w e to obtain that Z η e Z r0 sd−2 U (t) ds dt I2 ≍ V (δD (x)) (δD (x) + t + s)d 0 V (t) 0 38 IVAN BIOČIĆ ZORAN VONDRAČEK Z AND VANJA WAGNER e U(t) dt 0 V (t)(δD (x) + t) Z e V (δD (x)) η U (t) ≤ dt. δD (x) 0 V (t) (6.5) ≍ V (δD (x)) η Estimate of I3 : Lastly, note that for z ∈ D c \ B(x, r0 ) such that δD (z) < η, estimate (4.5) is equivalent to V (δD (x)) . PD (x, z) ≍ V (δDc (z)) Therefore, similarly as in the estimate of I1 we have Z e U (δDc (z)) I3 ≍ V (δD (x)) dz D3 V (δD c (z)) Z η e Z U (t) ≍ V (δD (x)) 1δD (w)=t dwdt 0 V (t) D c \B(x,r0 ) Z η e U (t) dt. ≍ V (δD (x)) 0 V (t) Since for t < η we have that δD (x) + t < diam(D) + η, it follows that I3  I2 . This proves that ! 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