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NONLOCAL ROBIN LAPLACIANS AND SOME REMARKS ON A PAPER BY FILONOV
, 2009
"... The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to selfadjoint Laplacians −∆Θ,Ω in L 2 (Ω; d n x) with (nonlocal and local) Robintype boundary conditions on bounded Lipschitz domains Ω ⊂ R n, n ∈ N, n ≥ 2. Second, we e ..."
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Cited by 16 (5 self)
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The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to selfadjoint Laplacians −∆Θ,Ω in L 2 (Ω; d n x) with (nonlocal and local) Robintype boundary conditions on bounded Lipschitz domains Ω ⊂ R n, n ∈ N, n ≥ 2. Second, we extend Friedlander’s inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains Ω, following an approach introduced by Filonov for this type of problems.
EIGENVALUE INEQUALITIES FOR MIXED STEKLOV PROBLEMS
"... Abstract. We extend some classical inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian to the context of mixed Steklov–Dirichlet and Steklov–Neumann eigenvalue problems. The latter one is also known as the sloshing problem, and has been actively studied for more than a centur ..."
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Cited by 10 (5 self)
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Abstract. We extend some classical inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian to the context of mixed Steklov–Dirichlet and Steklov–Neumann eigenvalue problems. The latter one is also known as the sloshing problem, and has been actively studied for more than a century due to its importance in hydrodynamics. The main results of the paper are applied to obtain certain geometric information about nodal sets of sloshing eigenfunctions. The key ideas of the proofs include domain monotonicity for eigenvalues of mixed Steklov problems, as well as an adaptation of Filonov’s method developed originally to compare the Dirichlet and Neumann eigenvalues. 1. Introduction and
Spectral boundary value problems and their linear operators
, 2009
"... The paper offers a selfconsistent account of the spectral boundary value problems developed from the perspective of general theory of linear operators in Hilbert spaces. An abstract form of spectral boundary value problem with generalized boundary condition is introduced and results on its solvabil ..."
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Cited by 7 (1 self)
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The paper offers a selfconsistent account of the spectral boundary value problems developed from the perspective of general theory of linear operators in Hilbert spaces. An abstract form of spectral boundary value problem with generalized boundary condition is introduced and results on its solvability complemented by representations of weak and strong solutions are obtained. The question of existence of a closed linear operator defined by a given boundary condition and description of its domain is studied in detail. This question is addressed on the basis of a version of Krein’s resolvent formula derived from the obtained representations for solutions. Usual resolvent identities for two operators associated with two different boundary conditions are written in terms of the so called Moperator and closed linear operators defining these conditions. Two examples illustrate the abstract core of the paper. Other applications to the theory of partial differential operators and to the mathematical physics are outlined.
Heat kernel estimates for pseudodifferential operators, fractional Laplacians and DirichlettoNeumann operators
 GENQIAN LIU
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