Grisvard's shift theorem near L^infinity and Yudovich theory on polygonal domains
Abstract
Let Omega be a bounded, simply connected domain with boundary of class C^{1,1} except at finitely many points S_j where the boundary is locally a corner of aperture alpha_j<=pi/2. Improving on results of Grisvard, we show that the solution Gf to the Dirichlet problem on Omega with data f in L^infinity(Omega) and homogeneous boundary conditions has exponentially integrable second derivatives. The proof uses sharp L^p bounds for singular integrals on power weighted spaces inspired by the work of Buckley. Our results allow for the extension of the Yudovich theory of existence, uniqueness and regularity of weak solutions to the Euler equations on Omega x (0,T) to polygonal domains Omega as above.
Keywords
Cite
@article{arxiv.1310.5444,
title = {Grisvard's shift theorem near L^infinity and Yudovich theory on polygonal domains},
author = {Francesco Di Plinio and Roger Temam},
journal= {arXiv preprint arXiv:1310.5444},
year = {2013}
}
Comments
20 pages; submitted. The authors are deeply grateful to Kabe Moen for reading an early version of the manuscript and for allowing the inclusion of his alternative proof of Proposition 5.2. The first named author wants to thank Stefan Steinerberger for stimulating discussions on the subject of this article