Parabolic equations with dynamical boundary conditions and source terms on interfaces

TL;DR

This paper studies parabolic equations with complex boundary conditions and inhomogeneities on interfaces, demonstrating maximal regularity under minimal assumptions, which aids in solving related nonlinear problems.

Contribution

It establishes maximal parabolic regularity for parabolic equations with mixed boundary conditions and interface inhomogeneities under minimal regularity assumptions.

Findings

Linear operator has maximal parabolic regularity in $L^p$-setting.

Results extend to nondegenerate quasilinear problems.

Applicable under minimal regularity assumptions.

Abstract

We consider parabolic equations with mixed boundary conditions and domain inhomogeneities supported on a lower dimensional hypersurface, enforcing a jump in the conormal derivative. Only minimal regularity assumptions on the domain and the coefficients are imposed. It is shown that the corresponding linear operator enjoys maximal parabolic regularity in a suitable $L^p$-setting. The linear results suffice to treat also the corresponding nondegenerate quasilinear problems.