Shape perturbation of a nonlinear mixed problem for the heat equation

TL;DR

This paper studies how the solution to a heat equation with nonlinear boundary conditions changes smoothly when the shape of an interior hole in the domain is slightly perturbed, using advanced mathematical tools.

Contribution

It proves the smooth dependence of solutions on the shape of the hole and establishes local uniqueness for the shape perturbations in a nonlinear heat problem.

Findings

Solution map is infinitely differentiable with respect to shape perturbations.

Solutions depend smoothly on the shape of the hole near the reference configuration.

The family of solutions is locally unique under small shape changes.

Abstract

We consider the heat equation in a domain that has a hole in its interior. We impose a Neumann condition on the exterior boundary and a nonlinear Robin condition on the boundary of the hole. The shape of the hole is determined by a suitable diffeomorphism $\phi$ defined on the boundary of a reference domain. Assuming that the problem has a solution $u_0$ when $\phi$ is the identity map, we demonstrate that a solution $u_\phi$ continues to exist for $\phi$ close to the identity map and that the "domain-to-solution" map $\phi\mapsto u_\phi$ is of class $C^\infty$. Moreover, we show that the family of solutions $\{u_\phi\}_{\phi}$ is, in a sense, locally unique. Our argument relies on tools from Potential Theory and the Implicit Function Theorem. Some remarks a the linear case complete the paper.