Grisvard's shift theorem near L^infinity and Yudovich theory on polygonal domains

TL;DR

This paper extends Grisvard's shift theorem to polygonal domains with corners, demonstrating that solutions to the Dirichlet problem with bounded data have exponentially integrable second derivatives, thereby broadening Yudovich's Euler solutions theory.

Contribution

It improves Grisvard's results by establishing exponential integrability of second derivatives for solutions on polygonal domains, enabling Yudovich's theory extension.

Findings

Solutions have exponentially integrable second derivatives.

Yudovich's Euler solutions theory extends to polygonal domains.

Sharp L^p bounds for singular integrals are used in proofs.

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.