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L. Fraenkel (2000)
An Introduction to Maximum Principles and Symmetry in Elliptic Problems
Sagun Chanillo, D. Grieser, K. Kurata (1999)
The Free Boundary Problem in the Optimization of Composite MembranesarXiv: Analysis of PDEs
Sagun Chanillo, D. Grieser, M. Imai, K. Kurata, I. Ohnishi (1999)
Symmetry Breaking and Other Phenomena in the Optimization of Eigenvalues for Composite MembranesCommunications in Mathematical Physics, 214
C. Pao (1993)
Nonlinear parabolic and elliptic equations
E. Harrell, Pawel Kröger, K. Kurata (2001)
On the Placement of an Obstacle or a Well so as to Optimize the Fundamental EigenvalueSIAM J. Math. Anal., 33
J. Lions (1971)
Optimal Control of Systems Governed by Partial Differential Equations
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.
Applied Mathematics and Optimization – Springer Journals
Published: Oct 1, 2004
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