Symmetry-Breaking Phenomena in an Optimization Problem for some Nonlinear Elliptic Equation
Symmetry-Breaking Phenomena in an Optimization Problem for some Nonlinear Elliptic Equation
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DOI：
10.1007/s00245-004-0803-5
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发表时间：
2004-08
影响因子：
1.8
通讯作者：
K. Kurata;Masataka Shibata;S. Sakamoto
中科院分区：
文献类型：
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作者：
K. Kurata;Masataka Shibata;S. Sakamoto
设是中的有界区域，具有Lipschitz边界，且当。设是一个可测子集，它属于规定的类。 存在唯一的全局最小值，我们用 函数\[\quad J_{\Omega，D}（v）=\frac12\int_{\Omega}|\nabla v| ^2\，dx+\frac{\lambda}{p+1}\int_{\Omega}| v| ^{p+1}\，dx -\int_{\Omega}\chi_Dv\，dx \] on。我们考虑优化问题，并说， a subsetwhich达到a optimal最优configuration配置to this problem问题. 本文证明了该优化问题在各种情况下最优构形的存在性、唯一性和非唯一性，以及保序和破序现象。
Letbe a bounded domain inwith Lipschitz boundary,andifandif. Letbe a measurable subset ofwhich belongs to the classfor the prescribedFor any, it is well known that there exists a unique global minimizer, which we denote by, 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. We consider the optimization problemand say that a subsetwhich attainsis 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 configurationto this optimization problem in various settings.