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 ϕ defined on the boundary of a reference domain. Assuming that the problem has a solution u0 when ϕ is the identity map, we demonstrate that a solution uϕ continues to exist for ϕ close to the identity map and that the “domain-to-solution” map ϕ↦uϕ is of class C∞. Moreover, we show that the family of solutions {uϕ}ϕ 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.