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Symmetry-Breaking Phenomena in an Optimization Problem for some Nonlinear Elliptic Equation.
- Source :
- Applied Mathematics & Optimization; Nov/Dec2004, Vol. 50 Issue 3, p259-278, 20p
- Publication Year :
- 2004
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- EQUATIONS
MATHEMATICAL optimization
MATHEMATICAL analysis
LIPSCHITZ spaces
SYMMETRY
Subjects
Details
- Language :
- English
- ISSN :
- 00954616
- Volume :
- 50
- Issue :
- 3
- Database :
- Complementary Index
- Journal :
- Applied Mathematics & Optimization
- Publication Type :
- Academic Journal
- Accession number :
- 14665360
- Full Text :
- https://doi.org/10.1007/s00245-004-0803-5