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 u_φ continues to exist for φ close to the identity map and that the "domain-to-solution" map φ↦ u_φ is of class C^∞. Moreover, we show that the family of solutions _φ\φ is, in a sense, locally unique. Our argument relies on tools from Potential Theory and the Implicit Function Theorem. Some remarks on the linear case complete the paper.