Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions

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Numerical minimization of eigenmodes of a membrane with respect to the domain (2004, unknown): “In [12] we could prove that the shape optimization problem”
Partially overdetermined elliptic boundary value problems (2008, methodology): “Then under our assumptions Lemma 1 of [ 19 ] remains true: it is enough to repeat the same proof, by taking care that the vector field V constructed therein is chosen with support contained in Γ . Then such lemma, combined with the simplicity of the first Dirichlet eigenvalue, shows that there exist”
Isoperimetric Inequalities for Eigenvalues of the Laplacian (2007, background): “It turns out that it is enough to hear the first two eigenfrequencies to determine whether the drum has the circular shape [6] In 1966, Mark Kac obtained the next term in the asymptotic behavior of ZΩ(t) [46].”
Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions (2016, unknown): “Generalizing the methods used in [25,26], take i, j ∈ 1, N with i = j , and define vi j := xi∂x j φ − x j∂xi φ in .”