Maximal regularity for non-autonomous Robin boundary conditions

TL;DR

This paper establishes $H$-maximal regularity for non-autonomous linear operators with Robin boundary conditions under a new Hölder continuity condition, enabling advanced analysis of nonlinear parabolic problems.

Contribution

It introduces a novel regularity condition on time-dependent forms, extending maximal regularity results to Robin boundary conditions.

Findings

Proves $H$-maximal regularity under Hölder continuity in interpolation spaces.

Applicable to nonlinear parabolic problems with Robin boundary conditions.

Enhances analytical tools for non-autonomous boundary value problems.

Abstract

We consider a non-autonomous Cauchy problem involving linear operators associated with time-dependent forms $a(t;.,.):V\times V\to {\mathbb{C}}$ where $V$ and $H$ are Hilbert spaces such that $V$ is continuously embedded in $H$. We prove $H$-maximal regularity under a new regularity condition on the form $a$ with respect to time; namely H{\"o}lder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.