The heat equation with rough boundary conditions and holomorphic functional calculus

TL;DR

This paper establishes bounded holomorphic functional calculus for the Laplace operator with rough boundary conditions on weighted spaces, leading to new regularity results for the heat equation with irregular boundary data.

Contribution

It extends the bounded $H^ abla$-calculus to weighted $L^p$-spaces beyond classical weights and characterizes the operator domain, impacting elliptic and parabolic regularity theory.

Findings

Bounded $H^ abla$-calculus on weighted $L^p$-spaces for non-$A_p$ weights.

Characterization of the Laplace operator's domain with rough boundary conditions.

New maximal regularity results for heat equations with irregular boundary data.

Abstract

In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded $H^\infty$-calculus on weighted $L^p$-spaces for power weights which fall outside the classical class of $A_p$-weights. Furthermore, we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity. In particular, we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data.