About Hölder-regularity of the convex shape minimizing λ2

Lamboley, Jimmy

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

Number

Publisher

Taylor and Francis

Pages

263-278

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