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Extremum Problems for Eigenvalues of Elliptic Operators
TLDR
The first eigenvalue of the Laplacian-Dirichlet operator was defined in this paper and the other Dirichlet eigenvalues were defined in this paper.Abstract:
Eigenvalues of elliptic operators.- Tools.- The first eigenvalue of the Laplacian-Dirichlet.- The second eigenvalue of the Laplacian-Dirichlet.- The other Dirichlet eigenvalues.- Functions of Dirichlet eigenvalues.- Other boundary conditions for the Laplacian.- Eigenvalues of Schrodinger operators.- Non-homogeneous strings and membranes.- Optimal conductivity.- The bi-Laplacian operator.read more
Citations
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Journal ArticleDOI
Geometrical structure of Laplacian eigenfunctions
TL;DR: The main focus is put onto multiple intricate relations between the shape of a domain and the geometrical structure of eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann or Robin boundary condition.
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Controllability of the discrete-spectrum Schrödinger equation driven by an external field
TL;DR: In this paper, the authors prove approximate controllability of the bilinear Schrodinger equation in the case of the uncontrolled Hamiltonian having a discrete non-resonant spectrum.
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Proof of the fundamental gap conjecture
Ben Andrews,Julie Clutterbuck +1 more
TL;DR: The fundamental gap conjecture of as discussed by the authors states that the difference between the first two Dirichlet eigenvalues (the spec- tral gap) of a Schrodinger operator with convex potential is bounded below by the spectral gap on an interval of the same diameter with zero potential.
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Minimization of the k -th eigenvalue of the Dirichlet Laplacian
TL;DR: In this article, the existence of a quasi-open set minimizing the k-th eigenvalue of the Dirichlet Laplacian among all sets of prescribed Lebesgue measure was proved.
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Generalizations and Properties of the Principal Eigenvalue of Elliptic Operators in Unbounded Domains
Henri Berestycki,Luca Rossi +1 more
TL;DR: In this article, the generalized principal eigenvalue of linear second-order elliptic operators in unbounded domains is derived and necessary and sufficient conditions for the validity of the maximum principle, as well as for the existence of positive eigenfunctions for the Dirichlet problem.
References
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Book
Isoperimetric inequalities in mathematical physics
George Pólya,Gábor Szegő +1 more
TL;DR: Isoperimetric Inequalities in Mathematical Physics (AM-27) as mentioned in this paper is an excellent survey of the literature in this area. But it is not a complete collection.
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Generalized isoperimetric inequalities
TL;DR: In this article, a generalized isoperimetric inequality for Green's functions for a potential which approaches zero at infinity is presented. But it is only for the case of a domain potential and not a general potential.