Shape analysis in Schauder spaces of the energy of heat problems in perturbed annular domains

This paper is devoted to the shape analysis of the energy of boundary value problems for the heat equation in a bounded perforated domain ${\mathrm{\Omega }}^o\setminus \overline{{\mathrm{\Omega }}^i[\phi ]}$ of ${\mathbb{R}}^n$, where the outer boundary is fixed, and the inner boundary is given by a $C^{1,\alpha }$-perturbation $\phi$ of the boundary of a reference cavity. Under standard Dirichlet or Neumann boundary conditions, we prove that, in a suitable neighborhood of the identity ${\phi }_0$, the domain-to-energy map is of class $C^{\mathrm{\infty }}$. The proof is based on the construction of a global diffeomorphism, smoothly depending on $\phi$, from the reference annulus onto the perturbed one, on a decomposition of the fixed domain into regions near, intermediate to, and far from the cavity, and on the smooth dependence of the layer heat potentials upon support perturbations.