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
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Article accepté pour publication ou publiéExternal document link
http://hal.archives-ouvertes.fr/hal-00530272/fr/Date
2011Journal name
Applicable AnalysisVolume
90Number
2Publisher
Taylor and Francis
Pages
263-278
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Lamboley, JimmyAbstract (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 boundariesRelated items
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