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

TL;DR

This paper constructs heat kernels for elliptic systems of linear elasticity with various boundary conditions, establishing Gaussian bounds under certain regularity and geometric conditions, and applies these results to Green's function construction.

Contribution

It develops a method to construct heat kernels for elliptic elasticity systems with boundary conditions, providing Gaussian bounds under minimal regularity assumptions.

Findings

Heat kernel exists under interior Hölder continuity.

Gaussian bounds hold if solutions are Hölder continuous up to boundary.

Green's function can be constructed in specific Lipschitz domains.

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\"older continuous in the interior. Moreover, we show that if weak solutions of the mixed problem are H\"older 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.