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

Technical Summary: Shape Analysis in Schauder Spaces of the Energy of Heat Problems in Perturbed Annular Domains

Problem Statement
This paper investigates the shape sensitivity of the energy associated with linear boundary value problems for the heat equation in bounded perforated domains of $\mathbb{R}^n$ ($n \geq 2$). Specifically, the authors consider a fixed outer domain $\Omega_o$ and an inner cavity $\Omega_i$ whose boundary is subjected to a $C^{1,\alpha}$-perturbation $\phi$. The domain of interest is the annular region $\Omega_o \setminus \Omega_i[\phi]$, where $\Omega_i[\phi]$ is the bounded component of the complement of the perturbed boundary $\phi(\partial \Omega_i)$.

The study focuses on two types of boundary conditions:

1. Dirichlet Problem: Prescribed temperature on both the outer boundary $\partial \Omega_o$ and the perturbed inner boundary $\phi(\partial \Omega_i)$.
2. Neumann Problem: Prescribed heat flux on both boundaries.

The primary objective is to establish the smoothness (specifically $C^\infty$ regularity) of the map that associates the shape perturbation parameter $\phi$ to the total energy of the solution, defined as $e_\phi(t) = \frac{1}{2} \int_{\Omega_o \setminus \Omega_i[\phi]} (u_\phi(t, y))^2 \, dy$.

Methodology
The authors employ the Functional Analytic Approach, a framework developed by Lanza de Cristoforis and collaborators, which relies heavily on potential-theoretic techniques and the implicit function theorem in Banach spaces. The methodology proceeds through several key stages:

1. Representation via Layer Potentials:

   • For the Dirichlet problem, the solution is represented as a sum of double layer heat potentials with densities defined on the boundaries. The authors provide an alternative representation to previous works (which used single layer potentials) by utilizing double layer potentials.
   • For the Neumann problem, the solution is represented using single layer heat potentials.
   • The boundary value problems are transformed into systems of integral equations on the boundaries.

2. Smoothness of Integral Operators:
   The analysis relies on the known smoothness of pullback layer heat potentials with respect to perturbations of the support (the boundary) and the density. The authors utilize results from Luzzini and Dalla Riva regarding the $C^\infty$ dependence of these operators on the diffeomorphism $\phi$.

3. Global Diffeomorphism Construction:
   A crucial technical step involves the construction of a global diffeomorphism $\Psi[\phi]$ that maps the reference fixed annulus $\Omega_o \setminus \Omega_i$ onto the perturbed annulus $\Omega_o \setminus \Omega_i[\phi]$. This map depends real-analytically on $\phi$ and is constructed by extending the boundary perturbation $\phi$ into the interior using a real-analytic extension operator, smoothly cut off to remain the identity near the outer boundary.

4. Decomposition of the Energy:
   To prove the smoothness of the energy map, the authors pull back the energy integral to the fixed reference domain using $\Psi[\phi]$. The resulting integral is decomposed into three regions relative to the inner cavity:

   • Near: A region close to the inner boundary where the diffeomorphism $\Psi[\phi]$ coincides with the extension of $\phi$.
   • Intermediate: A region where the diffeomorphism transitions.
   • Far: A region near the outer boundary where the diffeomorphism is the identity.
     This decomposition allows the authors to handle the singularities and dependencies of the layer potentials separately in each region, leveraging the smoothness of the pullback potentials and the regularity of the solution on fixed subdomains.

Key Contributions and Results

• Existence and Regularity of Solutions: The paper establishes the existence of a unique Schauder-classical solution for both the perturbed Dirichlet and Neumann problems in a neighborhood of the identity perturbation. It proves that the domain-to-solution map is of class $C^\infty$ when restricted to fixed subdomains compactly contained in the reference annulus.
• Main Theorem (Dirichlet Energy): The central result (Theorem 3.10) proves that the map $\phi \mapsto e_\phi$ is of class $C^\infty$ from a neighborhood of the identity in the space of diffeomorphisms to the space $C^1([0, T])$. The proof explicitly derives the time derivative of the energy and demonstrates its smooth dependence on $\phi$.
• Main Theorem (Neumann Energy): Analogous results are established for the Neumann problem (Theorem 4.4), confirming the $C^\infty$ smoothness of the energy map for heat flux boundary conditions.
• Alternative Representation: The authors provide a representation of caloric functions in annular domains using a sum of double layer potentials (Lemma 3.1), offering an alternative to the single-layer representations used in prior literature.

Significance and Claims
The paper claims novelty primarily in the shape analysis of the energy for parabolic problems using potential-theoretic techniques. While the smoothness of the solution map itself is a known result in the context of the Functional Analytic Approach (referencing works by Lanza de Cristoforis, Musolino, and others), the extension of this analysis to the energy functional is presented as a new contribution.

The authors note that while asymptotic behaviors of energy in elliptic homogenization problems have been studied, the smooth dependence of the energy of perturbed parabolic problems on the shape parameter has not been previously addressed via these methods. The work extends the applicability of the Functional Analytic Approach to time-dependent energy functionals, demonstrating that the energy map inherits the smoothness of the underlying solution and the geometric perturbation. The analysis is conducted in a framework that, while linear, is structured to allow for straightforward extension to nonlinear cases.

The paper does not propose new experimental applications or future numerical schemes but rather solidifies the theoretical foundation for shape optimization and sensitivity analysis in time-dependent heat transfer problems within perforated domains.