Minimal convex combinations of three sequential Laplace-Dirichlet eigenvalues. (English) Zbl 1305.49066

Summary: We study the shape optimization problem where the objective function is a convex combination of three sequential Laplace-Dirichlet eigenvalues. That is, for \(\alpha\geq 0, \beta\geq 0\), and \(\alpha+\beta\leq1\), we consider \(\inf\{\alpha\lambda_k(\Omega)+\beta\lambda_{k+1}(\Omega)+(1-\alpha-\beta)\lambda_{k+2}(\Omega):\Omega \;\text{open\;set\;in}\;\mathbb{R}^2 \;\text{and}\;|\Omega|\leq 1\}\). Here \(\lambda_k(\Omega)\) denotes the \(k\)-th Laplace-Dirichlet eigenvalue and \(|\cdot|\) denotes the Lebesgue measure. For \(k=1,2\), the minimal values and minimizers are computed explicitly when the set of admissible domains is restricted to the disjoint union of balls. For star-shaped domains, we show that for \(k=1\) and \(\alpha+2\beta\leq1\), the ball is a local minimum. For \(k=1,2\), several properties of minimizers are studied computationally, including uniqueness, connectivity, symmetry, and eigenvalue multiplicity.


49Q10 Optimization of shapes other than minimal surfaces
35J25 Boundary value problems for second-order elliptic equations


Full Text: DOI


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