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 a caloric family of Schauder functions defined on a bounded perforated domain $Ω^o \setminus \overline{Ω^i[ϕ]}$ of $\mathbb{R}^n$, where the outer boundary is fixed, and the inner boundary is obtained by a $C^{1,α}$-perturbation $ϕ$ of the boundary of a reference cavity $Ω^i$. Without imposing any boundary conditions, we prove that in a suitable neighborhood of the identity $ϕ_0$, the domain-to-energy map is of class $C^{\infty}$. The proof is based on the construction of a global diffeomorphism, smoothly depending on $ϕ$, from the reference annulus onto the perturbed one and on suitable regularity and smoothness assumptions on the pull-back family onto the reference domain. We then apply our main result to two boundary value problems: a nonlinear mixed Robin-type problem and a linear Dirichlet problem. After recalling some known existence and shape analysis results for the solutions, we prove that the corresponding domain-to-energy map is of class $C^{\infty}$. The proof is based on a decomposition of the fixed domain into near, intermediate, and far regions relative to the cavity, and on the smooth dependence of the layer heat potentials upon support perturbations.