Symmetry-Breaking Phenomena in an Optimization Problem for some Nonlinear Elliptic Equation |
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Authors: | Kazuhiro Kurata Masataka Shibata and Shigeru Sakamoto |
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Institution: | (1) Department of Mathematics, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji-shi, Tokyo 192-0397, Japan;(2) Department of Mathmatics, Tokyo Institue of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan;(3) 1-289-19 Ohnuma-cho, Kodaira-shi, Tokyo, Japan |
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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. |
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Keywords: | Symmetry-breaking phenomena Optimization Nonlinear elliptic problem |
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