Symmetry-Breaking Phenomena in an Optimization Problem for some Nonlinear Elliptic Equation

Let $Omega$ be a bounded domain in ${bf R^n}$ with Lipschitz boundary, $lambda >0,$ and $1le p le (n+2)/(n-2)$ if $nge 3$ and $1le p< +infty$if $n=1,2$. Let $D$ be a measurable subset of $Omega$ which belongs to the class${cal C}_{beta}={Dsubset Omega quad | quad |D|=beta}$for the prescribed $betain (0, |Omega|).$ For any $Din{cal C}_{beta}$, it is well known that there exists a unique global minimizer $uin H^1_0(Omega)$, which we denote by$u_D$, of the functional[quadJ_{Omega,D}(v)=frac12int_{Omega}|nabla v|^2, dx+frac{lambda}{p+1}int_{Omega}|v|^{p+1}, dx-int_{Omega}chi_Dv,dx ]on $H^1_0(Omega)$.We consider the optimization problem$E_{beta,Omega}=inf_{Din {cal C}_{beta}} J_D(u_D)$and say that a subset $D^*in {cal C}_{beta}$ which attains $E_{beta,Omega}$ is an optimal configuration to this problem. In this paper we show the existence, uniqueness and non-uniqueness, and symmetry-preserving and symmetry-breaking phenomena of theoptimal configuration $D^*$ to this optimization problem in various settings.