Symmetry breaking for a problem in optimal insulation

2016/01/09 by Dorin Bucur, Bucur, Dorin, Giuseppe Buttazzo +3 · 3 citations
Mathematics · #35B06 #35J25 #49J45 #49R05 #FOS: Mathematics #Optimization and Control (math.OC) #math.OC #msc:35B06 #msc:35J25 #msc:49J45 #msc:49R05

paper · pdf · doi:10.48550/arxiv.1601.02146

12 pages, 0 figures

arxiv created 2016/01/09 · arxiv updated 2016/01/12

Abstract

We consider the problem of optimally insulating a given domain Ω of 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 Ω to vary too.

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