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A Short Proof of Weyl's Lemma

✍ Scribed by Roland Lemmert

Publisher: John Wiley and Sons
Year: 1988
Tongue: English
Weight: 71 KB
Volume: 135
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.19881350119

No coin nor oath required. For personal study only.

✦ Synopsis

Let Q c R a be open and LZ~=ZcaD026 be n linear differential opentor with constant coefficients, and adjoint L*zi = C (l)'"'e,Dazs. We give D proof of the following variant of WEYL'S lemma. d Letit~rta. Every contiwma 8olutim of h = O in the distributional sense is the iocalhj uniform h i t of G+'-soh6.iOns u k , i.e., to euela bull Bfxo, rj with B ( q . T ) ~Q there exist.9 B seyumce (ur) of solutions converging uniformly to u MZ B(x0, r ) . --Proof. Let (F) be 8 WI&I sequence of test functions approxim&ing DIRAC'S function, for example, m is well known (and easy to prove), ukEG+', uk-u uniformly (eypecidly) on B b , 7).

NOW L(z)qIk (z-5 ) = L&pk (x -5 ) so that (by interchanging differentiation and

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