Heat kernel for the elliptic system of linear elasticity with boundary conditions

Justin Taylor, Seick Kim, Russell Brown

Research output: Contribution to journalArticle

Abstract

We consider the elliptic system of linear elasticity with bounded measurable coefficients in a domain where the second Korn inequality holds. We construct heat kernel of the system subject to Dirichlet, Neumann, or mixed boundary condition under the assumption that weak solutions of the elliptic system are Hölder continuous in the interior. Moreover, we show that if weak solutions of the mixed problem are Hölder continuous up to the boundary, then the corresponding heat kernel has a Gaussian bound. In particular, if the domain is a two dimensional Lipschitz domain satisfying a corkscrew or non-tangential accessibility condition on the set where we specify Dirichlet boundary condition, then we show that the heat kernel has a Gaussian bound. As an application, we construct Green's function for elliptic mixed problem in such a domain.

Original languageEnglish
Pages (from-to)2485-2519
Number of pages35
JournalJournal of Differential Equations
Volume257
Issue number7
DOIs
Publication statusPublished - 2014 Oct 1

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Linear Elasticity
Heat Kernel
Elliptic Systems
Elasticity
Mixed Problem
Boundary conditions
Weak Solution
Korn's Inequality
Lipschitz Domains
Mixed Boundary Conditions
Neumann Boundary Conditions
Accessibility
Green's function
Elliptic Problems
Dirichlet Boundary Conditions
Dirichlet
Interior
Coefficient
Hot Temperature

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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title = "Heat kernel for the elliptic system of linear elasticity with boundary conditions",
abstract = "We consider the elliptic system of linear elasticity with bounded measurable coefficients in a domain where the second Korn inequality holds. We construct heat kernel of the system subject to Dirichlet, Neumann, or mixed boundary condition under the assumption that weak solutions of the elliptic system are H{\"o}lder continuous in the interior. Moreover, we show that if weak solutions of the mixed problem are H{\"o}lder continuous up to the boundary, then the corresponding heat kernel has a Gaussian bound. In particular, if the domain is a two dimensional Lipschitz domain satisfying a corkscrew or non-tangential accessibility condition on the set where we specify Dirichlet boundary condition, then we show that the heat kernel has a Gaussian bound. As an application, we construct Green's function for elliptic mixed problem in such a domain.",
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Heat kernel for the elliptic system of linear elasticity with boundary conditions. / Taylor, Justin; Kim, Seick; Brown, Russell.

In: Journal of Differential Equations, Vol. 257, No. 7, 01.10.2014, p. 2485-2519.

Research output: Contribution to journalArticle

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AB - We consider the elliptic system of linear elasticity with bounded measurable coefficients in a domain where the second Korn inequality holds. We construct heat kernel of the system subject to Dirichlet, Neumann, or mixed boundary condition under the assumption that weak solutions of the elliptic system are Hölder continuous in the interior. Moreover, we show that if weak solutions of the mixed problem are Hölder continuous up to the boundary, then the corresponding heat kernel has a Gaussian bound. In particular, if the domain is a two dimensional Lipschitz domain satisfying a corkscrew or non-tangential accessibility condition on the set where we specify Dirichlet boundary condition, then we show that the heat kernel has a Gaussian bound. As an application, we construct Green's function for elliptic mixed problem in such a domain.

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