H\"older Regularity for Domains of Fractional Powers of Elliptic Operators with Mixed Boundary Conditions

TL;DR

This paper proves H"older regularity for solutions to elliptic PDEs with mixed boundary conditions on irregular domains, using classical techniques and minimal geometric assumptions, extending known results to fractional powers of operators.

Contribution

It establishes that fractional powers of elliptic operators inherit H"older regularity under minimal boundary interface conditions, simplifying previous proofs in space dimensions up to four.

Findings

Domains of fractional powers embed into H"older spaces under certain conditions.

The main premise for regularity is verified for irregular domains with minimal boundary assumptions.

The approach avoids Lipschitz charts, relying on measure-theoretic conditions.

Abstract

This work is about global H\"older regularity for solutions to elliptic partial differential equations subject to mixed boundary conditions on irregular domains. There are two main results. In the first, we show that if the domain of the realization of an elliptic differential operator in a negative Sobolev space with integrability $q > d$ embeds into a space of H\"older continuous functions, then so do the domains of suitable fractional powers of this operator. The second main result then establishes that the premise of the first is indeed satisfied. The proof goes along the classical techniques of localization, transformation and reflection which allows to fall back to the classical results of Ladyzhenskaya or Kinderlehrer. One of the main features of our approach is that we do not require Lipschitz charts for the Dirichlet boundary part, but only an intriguing metric/measure-theoretic condition on the interface of Dirichlet- and Neumann boundary parts. A similar condition was posed in a related work by ter Elst and Rehberg in 2015, but the present proof is much simpler, if only restricted to space dimension up to 4.