Calculus of Variations and Geometric Measure Theory
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J. Lamboley

About Hölder regularity of the convex shape minimizing $\lambda_2$

created by lamboley on 26 May 2016


Published Paper

Inserted: 26 may 2016
Last Updated: 26 may 2016

Journal: Applicable Analysis
Volume: 90
Number: 2
Pages: 263 - 278
Year: 2011


In this paper, we consider the well-known following shape optimization problem: \[\lambda_2(\Omega^*)=\min_{\stackrel{
=V_0} {\Omega\textrm{ convex}}} \lambda_2(\Omega),\] where $\lambda_2(\Omega)$ denotes the second eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions in $\Omega\subset\mathbb{R}^2$, and $
$ is the area of $\Omega$. We prove, under some technical assumptions, that any optimal shape $\Omega^*$ is $C^{1,\frac{1}{2}}$ 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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