Shape analysis in Schauder spaces of the energy of heat problems in perturbed annular domains
Abstract
This paper is devoted to the shape analysis of the energy of a caloric family of Schauder functions defined on a bounded perforated domain Ωo ∖ Ωi[ϕ]̅ of ℝn, where the outer boundary is fixed, and the inner boundary is obtained by a C1,α-perturbation ϕ of the boundary of a reference cavity Ωi. Without imposing any boundary conditions, we prove that in a suitable neighborhood of the identity ϕ0, the domain-to-energy map is of class C∞. 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∞. 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.
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BibTeX
@misc{persio2026shape,
title = {{Shape analysis in Schauder spaces of the energy of heat problems in perturbed annular domains}},
author = {Luca Di Persio and Riccardo Molinarolo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.00803},
eprint = {2607.00803},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.00803},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Shape analysis in Schauder spaces of the energy of heat problems in perturbed annular domains
AU - Luca Di Persio
AU - Riccardo Molinarolo
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2607.00803
UR - https://arxiv.org/abs/2607.00803
AB - 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 ∖ Ω^i[ϕ]̅ of 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^∞. 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^∞. 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.
ER -