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

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.

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