Solvability of the Dirichlet, Neumann and the regularity problems for parabolic equations with H\"older continuous coefficients

TL;DR

This paper proves the solvability of boundary value problems for parabolic equations with H"older continuous coefficients on Lipschitz domains, using layer potentials and Fredholm theory to establish invertibility and Rellich estimates.

Contribution

It introduces a new systematic approach for establishing invertibility of layer potentials for parabolic equations with H"older coefficients, enabling $L^2$-solvability results.

Findings

Established $L^2$-solvability of Dirichlet, Neumann, and regularity problems

Demonstrated invertibility of layer potentials using Fredholm theory

Developed parabolic Rellich-type estimates for these equations

Abstract

We establish the $L^2$-solvability of Dirichlet, Neumann and regularity problems for divergence-form heat (or diffusion) equations with H\"older-continuous diffusion coefficients, on bounded Lipschitz domains in $\mathbb{R}^n$. This is achieved through the demonstration of invertibility of the relevant layer-potentials which is in turn based on Fredholm theory and a new systematic approach which yields suitable parabolic Rellich-type estimates.