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Applicable Analysis
An International Journal
Volume 90, 2011 - Issue 2
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Original Articles

About Hölder-regularity of the convex shape minimizing λ2

Pages 263-278 | Received 03 Feb 2010, Accepted 22 May 2010, Published online: 04 Sep 2010
 

Abstract

In this article, we consider the well-known following shape optimization problem:

where λ2(Ω) denotes the second eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions in Ω ⊂ ℝ2, and |Ω| is the area of Ω. We prove, under some technical assumptions, that any optimal shape Ω* is and is not 𝒞1,α for any . 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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