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

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 $1\le p \le (n+2)/(n-2)$ if $n\ge 3$ and $1\le p< +\infty$ if $n=1,2$. Let $D$ be a measurable subset of $\Omega$ which belongs to the class ${\cal C}_{\beta}=\{D\subset \Omega \quad | \quad |D|=\beta\}$ for the prescribed $\beta\in (0, |\Omega|).$ For any $D\in{\cal C}_{\beta}$, it is well known that there exists a unique global minimizer $u\in H^1_0(\Omega)$, which we denote by $u_D$, of the functional $$\quad J_{\Omega,D}(v)=\frac12\int_{\Omega}|\nabla v|^2\, dx+\frac{\lambda}{p+1}\int_{\Omega}|v|^{p+1}\, dx -\int_{\Omega}\chi_Dv\,dx$$ on $H^1_0(\Omega)$. We consider the optimization problem $E_{\beta,\Omega}=\inf_{D\in {\cal C}_{\beta}} J_D(u_D)$ 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 the optimal configuration $D^*$ to this optimization problem in various settings. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

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

, Volume 50 (3) – Oct 1, 2004
20 pages

/lp/springer_journal/symmetry-breaking-phenomena-in-an-optimization-problem-for-some-BTgC0WUFHj
Publisher
Springer Journals
Subject
Mathematics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-004-0803-5
Publisher site
See Article on Publisher Site

### Abstract

Let $\Omega$ be a bounded domain in ${\bf R^n}$ with Lipschitz boundary, $\lambda >0,$ and $1\le p \le (n+2)/(n-2)$ if $n\ge 3$ and $1\le p< +\infty$ if $n=1,2$. Let $D$ be a measurable subset of $\Omega$ which belongs to the class ${\cal C}_{\beta}=\{D\subset \Omega \quad | \quad |D|=\beta\}$ for the prescribed $\beta\in (0, |\Omega|).$ For any $D\in{\cal C}_{\beta}$, it is well known that there exists a unique global minimizer $u\in H^1_0(\Omega)$, which we denote by $u_D$, of the functional $$\quad J_{\Omega,D}(v)=\frac12\int_{\Omega}|\nabla v|^2\, dx+\frac{\lambda}{p+1}\int_{\Omega}|v|^{p+1}\, dx -\int_{\Omega}\chi_Dv\,dx$$ on $H^1_0(\Omega)$. We consider the optimization problem $E_{\beta,\Omega}=\inf_{D\in {\cal C}_{\beta}} J_D(u_D)$ 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 the optimal configuration $D^*$ to this optimization problem in various settings.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Oct 1, 2004

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