Differential Equations.--Shape optimization for a nonlinear elliptic problem related to thermal insulation

• Published in: Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni Vol. 35; no. 1; p. 105
• Main Author: Barbato, Rosa
• Format: Journal Article
• Language: English
• Published: European Mathematical Society Publishing House 01.03.2024
• Subjects: Analysis; Differential equations
• ISSN: 1120-6330
• DOI: 10.4171/RLM/1035

In this paper, we consider a minimization problem of a nonlinear functional [Please download the PDF to view the mathematical expression] related to a thermal insulation problem with a convection term, where [OMEGA] is a bounded connected open set in [Please download the PDF to view the mathematical expression] and [Please download the PDF to view the mathematical expression] is a compact set. The Euler-Lagrange equation relative to [Please download the PDF to view the mathematical expression] is a p-Laplace equation, [Please download the PDF to view the mathematical expression], with a Robin boundary condition with parameter [beta] > 0. The main aim is to study extremum problems for [Please download the PDF to view the mathematical expression], among domains D with given geometrical constraints and [OMEGA] \ D of fixed thickness. In the planar case, we show that under perimeter constraint the disk maximizes [Please download the PDF to view the mathematical expression]. In the n -dimensional case we restrict our analysis to convex sets showing that the same is true for the ball but under different geometrical constraints. KEYWORDS.--Shape optimization, optimal insulation, mixed boundary conditions. MATHEMATICS SUBJECT CLASSIFICATION 2020. - 35J25 (primary); 49Q10 (secondary).