Shape perturbation of a nonlinear mixed problem for the heat equation

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 \(\phi \) $\varphi$ defined on the boundary of a reference domain. Assuming that the problem has a solution \(u_0\) $u_0$ when \(\phi \) $\varphi$ is the identity map, we demonstrate that a solution \(u_\phi \) $u_{\varphi }$ continues to exist for \(\phi \) $\varphi$ close to the identity map and that the “domain-to-solution” map \(\phi \mapsto u_\phi \) $\varphi \mapsto u_{\varphi }$ is of class \(C^\infty \) $C^{\infty }$ . Moreover, we show that the family of solutions \(\{u_\phi \}_{\phi }\) ${\{u_{\varphi }\}}_{\varphi }$ is, in a sense, locally unique. Our argument relies on tools from Potential Theory and the Implicit Function Theorem. Some remarks on a linear case complete the paper.