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About Hölder-regularity of the convex shape minimizing λ2

Lamboley, Jimmy (2011), About Hölder-regularity of the convex shape minimizing λ2, Applicable Analysis, 90, 2, p. 263-278. http://dx.doi.org/10.1080/00036811.2010.496361

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00530272/fr/
Date
2011
Journal name
Applicable Analysis
Volume
90
Number
2
Publisher
Taylor and Francis
Pages
263-278
Publication identifier
http://dx.doi.org/10.1080/00036811.2010.496361
Metadata
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Author(s)
Lamboley, Jimmy
Abstract (EN)
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.
Subjects / Keywords
overdetermined boundary value problems; convex constraint; eigenvalues of the Laplacian; shape optimization; conformal map; regularity of free boundaries

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