Calculus of Variations and Geometric Measure Theory

D. Bucur - G. Buttazzo - C. Nitsch

Symmetry breaking for a problem in optimal insulation

created by buttazzo on 09 Jan 2016
modified by bucur on 23 Jan 2018


Accepted Paper

Inserted: 9 jan 2016
Last Updated: 23 jan 2018

Journal: Journal de Mathématiques Pures et Appliquées
Year: 2017


We consider the problem of optimally insulating a given domain $\Omega$ of ${\mathbb{R}}^d$; this amounts to solve a nonlinear variational problem, where the optimal thickness of the insulator is obtained as the boundary trace of the solution. We deal with two different criteria of optimization: the first one consists in the minimization of the total energy of the system, while the second one involves the first eigenvalue of the related differential operator. Surprisingly, the second optimization problem presents a symmetry breaking in the sense that for a ball the optimal thickness is nonsymmetric when the total amount of insulator is small enough. In the last section we discuss the shape optimization problem which is obtained letting $\Omega$ to vary too.

Keywords: optimal insulation, symmetry breaking, Robin boundary conditions