By Antoine Henrot

Difficulties linking the form of a website or the coefficients of an elliptic operator to the series of its eigenvalues are one of the such a lot attention-grabbing of mathematical research. during this e-book, we concentrate on extremal difficulties. for example, we glance for a site which minimizes or maximizes a given eigenvalue of the Laplace operator with quite a few boundary stipulations and diverse geometric constraints. We additionally ponder the case of services of eigenvalues. We examine comparable questions for different elliptic operators, similar to the Schrödinger operator, non homogeneous membranes, or the bi-Laplacian, and we glance at optimum composites and optimum insulation difficulties when it comes to eigenvalues.Providing additionally a self-contained presentation of classical isoperimetric inequalities for eigenvalues and 30 open difficulties, this e-book could be necessary for natural and utilized mathematicians, relatively these drawn to partial differential equations, the calculus of adaptations, differential geometry, or spectral conception.

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Let uk , k ≥ 2 be an eigenfunction of the elliptic operator L with Dirichlet or Neumann boundary conditions. The connected components of the open sets Ω+ = {x ∈ Ω, uk (x) > 0} and Ω− = {x ∈ Ω, uk (x) < 0} are called the nodal domains of uk . 2 (Nodal domains). Let uk , k ≥ 2 be the k-th eigenfunction of the elliptic operator L with Dirichlet or Neumann boundary conditions. Then, uk has at most k nodal domains. g. [58]. 5) without regularity assumptions. 4. Perforated domains 15 also implies that the ﬁrst eigenfunction must be simple in the connected case since two non negative and non zero functions cannot be orthogonal.

Mosco, G. Buttazzo and the Italian school, D. P. Zolesio, D. Bucur and the French school, V. Sverak, D. Daners, W. Arendt,... among others. For a complete study of this topic, we refer to [44], [104]. γ-convergence Let us begin with the deﬁnition of γ-convergence (for the Laplacian). 9. Let D be a ﬁxed ball, Ωn ⊂ D a sequence of open sets and Ω ⊂ D γ an open set. 6) on Ωn with right-hand side f converges (strongly) in L2 (D) to ufΩ , the solution on Ω (as usual, every function in H01 (Ωn ) is extended by zero outside Ωn ).

Then, k+n k+n λi (Ω) = min i=k+1 Ω i=k+1 |∇vi (x)|2 dx where (vi ) is an orthonormal family in L2 (Ω) satisfying 1, 2, . . , k. 38) vi uj dx = 0, j = k+n i=k+1 where (vi ) is a family in H01 (Ω) satisfying 0, j = 1, 2, . . , k. 2 Monotonicity Let us consider two open bounded sets such that Ω1 ⊂ Ω2 . This inclusion induces a natural embedding H01 (Ω1 ) → H01 (Ω2 ) just by extending by zero functions in H01 (Ω1 ). 40) (since the minimum is taken over a larger class for λD k (L, Ω2 )). Moreover, the inequality is strict as soon as Ω2 \ Ω1 contains a set of positive capacity (since the ﬁrst eigenfunction cannot vanish on such a set).