Regularity of Solutions of Linear Second Order Elliptic and Parabolic Boundary Value Problems on Lipschitz Domains

TL;DR

This paper proves that solutions to certain elliptic and parabolic boundary value problems on Lipschitz domains are Hölder continuous and well-posed, given specific regularity conditions on the data and boundary conditions.

Contribution

It establishes Hölder continuity of solutions for elliptic problems with Robin boundary conditions and demonstrates well-posedness of related parabolic problems on Lipschitz domains.

Findings

Solutions are Hölder continuous with sufficiently regular right-hand sides.

Parabolic problems with Robin or Wentzell-Robin boundary conditions are well-posed.

Results apply to elliptic operators with bounded measurable coefficients.

Abstract

For a linear, strictly elliptic second order differential operator in divergence form with bounded, measurable coefficients on a Lipschitz domain $\Omega$ we show that solutions of the corresponding elliptic problem with Robin and thus in particular with Neumann boundary conditions are Hoelder continuous for sufficiently $L^p$-regular right-hand sides. From this we deduce that the parabolic problem with Robin or Wentzell-Robin boundary conditions are well-posed on $\mathrm{C}(\bar{\Omega})$.