Regularity of the optimal shape for the first eigenvalue of the laplacian with volume and inclusion constraints

By Tanguy Brianccon, Jimmy Lamboley. 2008

68 citations recorded by the source corpus.

Abstract We consider the well-known following shape optimization problem: λ 1 ( Ω ∗ ) = min | Ω | = a Ω ⊂ D λ 1 ( Ω ) , where λ 1 denotes the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary condition, and D is an open bounded set (a box). It is well-known that the solution of this problem is the ball of volume a if such a ball exists in the box D (Faber–Krahn's theorem). In this paper, we prove regularity properties of the boundary of the optimal shapes Ω ∗ in any case and in any dimension. Full regularity is obtained in dimension 2.

Subfields: Applied Mathematics, Civil and Structural Engineering, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/c662205512e0b573357fde75ecfa519ba85e005b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Civil and Structural Engineering

In Computational Theory and Mathematics

Citations in the corpus

Regularity of the optimal sets for some spectral functionals (2016, background): “The only known result prior to the present paper concerning the regularity of the free boundary of the optimal sets is due to Briançon and Lamboley [5] who prove that the optimal sets for the problem min { λ1(Ω) : Ω ⊂ D open, |Ω| = 1 } , (1.3) in a bounded open set D ⊂ Rd have smooth boundary up to”
Faber–Krahn inequalities in sharp quantitative form (2013, methodology): “Since we are now facing a perturbed free boundary type problem, we aim to apply the techniques of Alt and Caffarelli [3] (see also [12, 13]) to show the regularity of ∂Uj = ∂{uj > 0} and to obtain the smooth convergence of Uj to B1.”