$\mathrm{L}_p$-maximal regularity for parabolic and elliptic boundary value problems with boundary conditions of mixed differentiability orders

TL;DR

This paper extends the theory of $ ext{L}_p$-maximal regularity for parabolic and elliptic boundary value problems to include mixed boundary conditions with different differentiability orders, broadening previous homogeneous boundary condition results.

Contribution

It generalizes existing $ ext{L}_p$-maximal regularity results to boundary conditions with mixed differentiability orders, filling a notable gap in the theory.

Findings

Extended $ ext{L}_p$-maximal regularity to mixed boundary conditions.

Adapted existing strategies to more general boundary conditions.

Provided a framework for future research on complex boundary value problems.

Abstract

In the theory of non-linear parabolic and elliptic partial differential equations, the notion of maximal regularity plays an essential role in establishing existence, regularity and boundedness of solutions. There is a long history of works where sufficient conditions for maximal regularity have been established: First scalar equations and systems of finitely many coupled equations have been considered. Around 2000, the vector-valued case with infinite-dimensional range space $E$ became accessible to the development and progress in theory of $\mathcal{R}$-bounded operator families and its close connection to the $\mathcal{H}^\infty$-calculus. The ground-braking results by Denk, Hieber and Pr\"uss for $\mathrm{L}_p$-maximal regularity of vector-valued parabolic and elliptic boundary value problems, however, were restricted to boundary conditions with homogeneous principle parts of the boundary symbol, in contrast to some previous results by Ladyszenskaya, Solonnikov and Uralceva for finite-component systems which also allow for, e.g. both Dirichlet and (mixed) flux boundary conditions at the same position. In this manuscript we aim for closing this gap, and extend the results of Denk, Hieber and Pr\"uss to this slightly more general situation. To this end, we closely review the strategy used in their works and adapt it to the situation considered here.