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∞-calculus on weighted Lp-spaces for power weights which fall outside the classical class of Ap-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.
Highlights
Often solutions to PDEs can have blow-up behavior near the boundary of an underlying domain O ⊆ Rd
Using weighted spaces with weights of the form wγO (x) := dist(x, ∂O)γ for appropriate values of γ, allows for additional flexibility and even obtain well-posedness for problems which appear ill-posed at first sight
The H ∞-functional calculus properties of differential operators on weighted space have been treated in several papers as well
Summary
Often solutions to PDEs can have blow-up behavior near the boundary of an underlying domain O ⊆ Rd. Dirichlet boundary conditions Dir for bounded C2-domains O This operator and its generalizations have been studied in many papers Dir and its functional calculus on weighted spaces which do not fall into the classical setting, but which are useful for certain partial differential equations. As a further application we will derive a maximal regularity result for the heat equation on weighted spaces with rough inhomogeneous boundary conditions. In ours proofs the main technical reason that we can extend the range of γ ’s in the Dirichlet setting is that the heat kernel on a half space has a zero of order one at the boundary. This is mainly because it can be convenient to write Sobolev spaces as the intersection of several simpler vector-valued Sobolev spaces
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