The heat equation with rough boundary conditions and holomorphic functional calculus

Nick Lindemulder; Mark Veraar

doi:10.1016/j.jde.2020.04.023

The heat equation with rough boundary conditions and holomorphic functional calculus

Nick Lindemulder, Mark Veraar^*

^*Corresponding author for this work

Analysis

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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 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.

Original language	English
Pages (from-to)	5832-5899
Number of pages	68
Journal	Journal of Differential Equations
Volume	269
Issue number	7
DOIs	https://doi.org/10.1016/j.jde.2020.04.023
Publication status	Published - 2020

Keywords

Functional calculus of the Laplace operator
Heat equation with inhomogeneous Dirichlet boundary conditions
Maximal regularity
Mixed-norms
Traces
Weights

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10.1016/j.jde.2020.04.023

1-s2.0-S0022039620301947-mainFinal published version, 672 KBLicence: CC BY

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title = "The heat equation with rough boundary conditions and holomorphic functional calculus",

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.",

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T1 - The heat equation with rough boundary conditions and holomorphic functional calculus

AU - Lindemulder, Nick

AU - Veraar, Mark

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AB - 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.

KW - Functional calculus of the Laplace operator

KW - Heat equation with inhomogeneous Dirichlet boundary conditions

KW - Maximal regularity

KW - Mixed-norms

KW - Traces

KW - Weights

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DO - 10.1016/j.jde.2020.04.023

M3 - Article

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SN - 0022-0396

VL - 269

SP - 5832

EP - 5899

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 7

ER -

The heat equation with rough boundary conditions and holomorphic functional calculus

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