About Hölder-regularity of the convex shape minimizing λ2

Details About Hölder-regularity of the convex shape minimizing λ2 About Hölder-regularity of the convex shape minimizing λ2

2010-10-29

arxiv.org/abs/1010.6239v1

Applicable Analysis (2010) 1

Mathematics

Optimization and Control

Scientific paper

In this paper, we consider the well-known following shape optimization problem: $$\lambda_2(\Omega^*)=\min_{\stackrel{|\Omega|=V_0} {\Omega\textrm{ convex}}} \lambda_2(\Omega),$$ where $\lambda_2(\Om)$ denotes the second eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions in $\Om\subset\R^2$, and $|\Om|$ is the area of $\Om$. We prove, under some technical assumptions, that any optimal shape $\Omega^*$ is $\mathcal{C}^{1,\frac{1}{2}}$ and is not $\C^{1,\alpha}$ for any $\alpha>\frac{1}{2}$. We also derive from our strategy some more general regularity results, in the framework of partially overdetermined boundary value problems, and we apply these results to some other shape optimization problems.

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