$L^p$-estimates for a transmission problem of mixed elliptic-parabolic type

TL;DR

This paper establishes $L^p$-estimates for solutions to a coupled elliptic-parabolic transmission problem, addressing discontinuous coefficients in bounded domains using Fourier multiplier techniques.

Contribution

It introduces novel $L^p$-estimates for elliptic-parabolic transmission problems with discontinuous coefficients in bounded domains.

Findings

Proves a priori estimates for strong solutions in $L^p$-Sobolev spaces.

Handles elliptic-parabolic equations with discontinuous coefficients.

Utilizes Fourier multiplier techniques for analysis.

Abstract

We consider the situation when an elliptic problem in a subdomain $\Omega_1$ of an $n$-dimensional bounded domain $\Omega$ is coupled via inhomogeneous canonical transmission conditions to a parabolic problem in $\Omega\setminus\Omega_1$. In particular, we can treat elliptic-parabolic equations in bounded domains with discontinuous coefficients. Using Fourier multiplier techniques, we prove an a priori estimate for strong solutions to the equations in $L^p$-Sobolev spaces.