Shape optimization for a nonlinear elliptic problem related to thermal insulation

In this paper, we consider a minimization problem of a nonlinear functional Iβ,p(D,Ω) related to a thermal insulation problem with a convection term, where Ω is a bounded connected open set in ⁿ and D⊂Ω̄ is a compact set. The Euler–Lagrange equation relative to Iβ,p is a p -Laplace equation, 1<p<∞ , with a Robin boundary condition with parameter β>0 . The main aim is to study extremum problems for Iβ,p(D,Ω) , among domains D with given geometrical constraints and Ω D of fixed thickness. In the planar case, we show that under perimeter constraint the disk maximizes Iβ,p . 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.