  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.
