发表机构
Università degli Studi di Verona（维罗纳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有界穿孔域中热方程边值问题能量关于内边界C^{1,α}扰动的光滑依赖性，证明在恒等映射附近域到能量映射为C∞类。
AI 中文摘要
本文致力于研究$\mathbb{R}^n$中有界穿孔域$\Omega^o \setminus \overline{\Omega^i[\phi]}$中热方程边值问题能量的形状分析，其中外边界固定，内边界由参考空腔边界的$C^{1,\alpha}$-扰动$\phi$给出。在标准Dirichlet或Neumann边界条件下，我们证明在恒等映射$\phi_0$的适当邻域内，域到能量映射属于$C^{\infty}$类。证明基于构造一个光滑依赖于$\phi$的从参考环到扰动环的整体微分同胚，将固定域分解为靠近空腔、中间区域和远离空腔的区域，以及层热势对支撑扰动的光滑依赖性。
英文摘要
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.
