Nothing Special   »   [go: up one dir, main page]
Scribd Logo
Upload

Welcome to Scribd!

  • 18 views

    Jr. FR

    Uploaded by

    Ola
    This document provides definitions and context for proving a regularity theorem for elliptic partial differential equations. It defines key terms like elliptic operators, characteristic polynomials, and Sobolev spaces. The main result is Theorem 8.12 which will show that solutions to elliptic PDEs have greater smoothness than the PDE alone implies. Context is provided on the Laplace equation to motivate investigating regularity of solutions to PDEs.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Jr. FR

    Uploaded by

    Ola
    18 views2 pages
    This document provides definitions and context for proving a regularity theorem for elliptic partial differential equations. It defines key terms like elliptic operators, characteristic polynomials, and Sobolev spaces. The main result is Theorem 8.12 which will show that solutions to elliptic PDEs have greater smoothness than the PDE alone implies. Context is provided on the Laplace equation to motivate investigating regularity of solutions to PDEs.

    Original Description:

    Original Title

    3

    Copyright

    © © All Rights Reserved

    Available Formats

    PDF, TXT or read online from Scribd

    Did you find this document useful?

    Is this content inappropriate?

    This document provides definitions and context for proving a regularity theorem for elliptic partial differential equations. It defines key terms like elliptic operators, characteristic polynomials, and Sobolev spaces. The main result is Theorem 8.12 which will show that solutions to elliptic PDEs have greater smoothness than the PDE alone implies. Context is provided on the Laplace equation to motivate investigating regularity of solutions to PDEs.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    18 views2 pages

    Jr. FR

    Uploaded by

    Ola
    This document provides definitions and context for proving a regularity theorem for elliptic partial differential equations. It defines key terms like elliptic operators, characteristic polynomials, and Sobolev spaces. The main result is Theorem 8.12 which will show that solutions to elliptic PDEs have greater smoothness than the PDE alone implies. Context is provided on the Laplace equation to motivate investigating regularity of solutions to PDEs.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

    of the proof consists in showing that this functional is continuous, i.e.

    ,
    that there is a distribution u E fi.&'(R") that satisfies
    (10) u(P(D)¢) = ¢(0) (¢ E fi.&(R")),
    because then the distribution E = u satisfies

    (P(D)E)(¢) = E(P( - D)¢) = u((P( - D)¢) V )


    = u(P(D)({J) = ({J(O) = ¢(0) = b(¢),
    so that P(D)E {>, as desired.
    =
    •

    Lemma 8.3, applied to P¢ t/1, yields


    A

    (1 1) I cj;(t) l < Ar - N l I �(t + rw) I da.(w) (t E R").


    Jr.
    By the inversion theorem, ¢(0) fR• ¢ dmn . Thus ( 1 1), (2), and (9) give
    =

    (12) I ¢(0) I < Ar - N II P(D)¢ 11 (¢ E fi.&(R")).


    Let Y be the subspace of fi.&(R") that consists of the functions
    P(D)¢, ¢ E fi.&(R"). By (12), the Hahn-Banach theorem 3.3 shows that
    the linear functional that is defined on Y by P(D)¢ -+ ¢(0) extends to a
    linear functional u on E!&(R") that satisfies (10) as well as
    ( 1 3) (t/1 E fi.&(R")).
    By (3), u E fi.&'(R"). This completes the proof. Ill/
    Elliptic Equations
    8.6 Introduction If u is a twice continuously differentiable function in
    some open set n c R 2 that satisfies the Laplace equation
    82u 82u
    (1) 0,
    ax 2 + 8y 2
    =

    then it is very well known that u is actually in C00(Q), simply because every
    real harmonic function in n is (locally) the real part of a holomorphic func­
    tion. Any theorem of this type--one which asserts that every solution of a
    certain differential equation has stronger smoothness properties than is a
    priori evident-is called a regularity theorem.
    We shall give a proof of a rather general regularity theorem for elliptic
    partial differential equations. The term " elliptic " will be defined presently.
    It may be of interest to see, first of all, that the equation
    82 u
    (2) 0
    ax ay
    =

    behaves quite differently from (1), since it is satisfied by every function u of


    the form u(x, y) = f(y), where f is any differentiable function. In fact, if (2) is

    ( )
    interpreted to mean
    j_ au
    (3) = 0,
    ay ax
    then / can be a perfectly arbitrary function.

    8.7 Definitions Suppose n is open in R ", N is a positive integer,


    fa E C00(!l) for every multi-index a. with I a. I < N, and at least one fa with
    I a. I = N is not identically 0. These data determine a linear differential oper­
    ator
    (1 )

    which acts on distributions u E .@'(Q) by


    (2) Lu L fa Da u .
    I a I ,;; N
    =

    The order of L is N. The operator


    (3) L fa Da
    lai = N
    is the principal part of L. The characteristic polynomial of L is
    (4) p(x, y) = L fa(x)ya (x E Q, y E R").
    l ai = N
    This is a homogeneous polynomial of degree N in the variables y =
    (y l , . . . , y.), with coefficients in C00(Q).
    The operator L is said to be elliptic if p(x, y) # 0 for every x E n and
    for every y E R ", except, of course, for y = 0. Note that ellipticity is defined
    in terms of the principal part of L; the lower-order terms that appear in (1)
    play no role.
    For example, the characteristic polynomial of the Laplacian

    (5)

    - A
    is p(x, y) = (yf + · · · + y;), so that is elliptic.
    On the other hand, if L = a 2jax 1 ax 2 , then p(x, y) = - y 1 y 2 , and L is
    not elliptic.
    The main result that we are aiming at (Theorem 8.12) involves some
    special spaces of tempered distributions, which we now describe.

    8.8 Sobolev spaces Associate to each real number s a positive measure


    f.ls on R " by setting

    (1) df.15(y) = (1 + I y 1 2)5 dm.(y).

    You might also like