非线性Robin边值问题的最优绝缘与浓度破碎

中文摘要 AI 辅助

我们考虑由$p$-Laplace算子（$p>1$）驱动的$\mathbb{R}^N$中有界域的最优绝缘问题。我们对物体与环境之间的对流热传递进行建模，这在绝缘前对应于非线性Robin边值问题。假设物体被厚度为$\varepsilon^{\frac{1}{p-1}}$的薄层绝缘材料包围，我们计算了当$\varepsilon \to 0^+$时控制能量泛函的$\Gamma$-极限。此外，我们研究了在绝缘材料总质量固定下，所有可能分布中热含量的优化。最后，我们揭示了一个浓度破碎现象。在适当的非退化条件下，如果域的边界是连通的或外部温度分布是常数，那么当总质量足够小时，最优绝缘层无法覆盖整个边界。这被证明是最优的：一个显式例子表明，不连通的边界可能引发异常的双相变，导致绝缘即使在中间质量范围内也会再次破裂。

英文摘要

We consider an optimal insulation problem for a bounded domain in $\mathbb{R}^N$ driven by the $p$-Laplace operator ($p>1$). We model the convective heat transfer between the body and the environment, which corresponds, before insulation, to a nonlinear Robin boundary value problem. Assuming the body is surrounded by a thin layer of insulating material of size $\varepsilon^{\frac{1}{p-1}}$, we compute the $Γ$-limit of the governing energy functional as $\varepsilon \to 0^+$. Furthermore, we study the optimization of the heat content among all possible distributions of the insulating material with a fixed total mass. Finally, we highlight a concentration breaking phenomenon. Under a suitable non-degeneracy condition, if the boundary of the domain is connected or the external temperature profile is constant, the optimal insulating layer fails to cover the entire boundary whenever the total mass is sufficiently small. This is shown to be optimal: an explicit example provides that a disconnected boundary can trigger an anomalous double-phase transition, causing the insulation to fracture again even at intermediate mass regimes.

发表机构

• Università degli studi di Napoli Federico II（那不勒斯费德里科二世大学）
• “Sapienza” Università di Roma（罗马大学）

机构由 AI 辅助整理，请以论文原文为准。