On the isoperimetric and isodiametric inequalities and the minimisation of eigenvalues of the Laplacian

2023/08/23 by Farrington, Sam
#35J25 #35P15 #49Q10 #49R05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2308.12245

Abstract

We consider the problem of minimising the k-th eigenvalue of the Laplacian with some prescribed boundary condition over collections of convex domains of prescribed perimeter or diameter. It is known that these minimisation problems are well-posed for Dirichlet eigenvalues in any dimension d≥ 2 and any sequence of minimisers converges to the ball of unit perimeter or diameter respectively as k→ +∞. In this paper, we show that the same is true in the case of Neumann eigenvalues under diameter constraint in any dimension and under perimeter constraint in dimension d=2. We also consider these problems for mixed Dirichlet-Neumann eigenvalues, under an additional geometric constraint, and discuss some applications of our proof techniques.

Related

• Optimization of Neumann Eigenvalues under convexity and geometric constraints
• Geometric inequalities between Dirichlet and Neumann eigenvalues
• Continuity of eigenvalues and shape optimisation for Laplace and Steklov\n problems
• Minimization of the eigenvalues of the dirichlet-laplacian with a diameter constraint
• Optimal bounds for Neumann eigenvalues in terms of the diameter