Extremum Problems for Eigenvalues of Elliptic Operators

By Antoine Henrot. 2006

970 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/931a8d58bfef2ea68d7f116d1147ca3cfbd310d9

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Spectral optimization for the Stekloff--Laplacian: the stability issue (2012, background): “These are shape optimization problems where the functional to be optimized is a function of the spectrum of an elliptic operator, typically the Laplacian −∆: the prototypical case is when this functional coincides with a single eigenvalue of the operator (see the book [12] or the recent survey paper”
Existence of minimizers for spectral problems (2011, background): “The following is a very useful characterization of the eigenvalues, which we will need later, and whose proof can be found for instance in [8].”
Minimization of the first eigenvalue in problems involving the bi-laplacian (2008, background): “For a collection of results on eigenvalue problems in the linear case we refer to the recent book [13] and the references therein.”
Isoperimetric Inequalities for Eigenvalues of the Laplacian (2007, background): “Other interesting open problems (and much else of interest) can be found in the forthcoming book by Henrot [83].”
Geometrical Structure of Laplacian Eigenfunctions (2012, background): “, φm (said to be in the Sobolev space H (1)(Ω)) [138, 238].”
Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions (2016, neutral): “We refer for instance to [18, 32, 33, 34, 35, 36] where Dirichlet boundary conditions are considered, and to [24, Chapter 9] for a survey on these problems.”
Homogeneous trees of second order Sturm-Liouville equations: A general theory and program (2008, background): “In higher dimensions, this becomes an upper bound (and not a sharp one) for the number of nodal domains [55, 119].”
Domain Perturbation for Linear and Semi-Linear Boundary Value Problems (2008, neutral): “As the measure goes to zero λ1(Bn) → ∞, and by the isoperimetric inequality for the first eigenvalue of the Dirichlet problem (see [20,83]) we get λ1( n ∩ ̄c) ≥ cλ1(Bn) → ∞.”
Minimizing Neumann fundamental tones of triangles: An optimal Poincaré inequality (2009, background): “For broad surveys of isoperimetric eigenvalue inequalities, see the paper by Ashbaugh [3], and the monographs of Bandle [5], Henrot [13], Kesavan [14] and Pólya–Szegő [21].”
Generalizations and Properties of the Principal Eigenvalue of Elliptic Operators in Unbounded Domains (2010, neutral): “3 in [19] for other continuity results for self-adjoint operators in bounded domains obtained via γ-convergence.”
Maximization of the second non-trivial Neumann eigenvalue (2018, methodology): “The structure of these N -functions is inspired by the functions built by Weinberger, that we briefly describe below (see, for instance, [12], [21] for more details).”