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Spectral theory, clustering problems and differential equations on metric graphs
Publication . Hofmann, Matthias; Kennedy, James Bernard; Tavares, Hugo Ricardo Nabais
In the first part we prove a general existence theory for constrained minimization problems for functionals defined on function spaces on metric measure spaces (M, d, μ). We apply this theory to functionals defined on metric graphs G. We show how the existence of solutions can be obtained via decomposition methods using spectral properties of the operator A associated with the form a(・, ・) and discuss the spectral quantities involved. Concrete examples considered include higher order NLS functionals and metric graphs with infinite edge set and magnetic potentials. This generalizes results obtained by Adami, Serra and Tilli [JFA 271 (2016), 201- 223], and Cacciapuoti, Finco and Noja [Nonlinearity 30 (2017), 3271-3303], among others.
In the second part we consider spectral minimal partitions of compact metric graphs. We motivate their study through Nehari ground state problems and certain penalized systems. We relate a class of minimal partitions to eigenvalues of the Laplacian and show sharp lower and upper estimates for the associated spectral minimal energies LDk ,∞ and LNk ,∞, estimates between these energies and eigenvalues of the Laplacian, which in some cases result in better estimates than the ones previously obtained in Berkolaiko et al [J. Phys. A 50 (2017), 365201] In the third partwe establish metric graph counterparts of Pleijel’s theorem on the asymptotics of the number of nodal domains νn of the n-th eigenfunction(s) of a broad class of operators on compact metric graphs. Among other things, these results characterize the accumulation points of the sequence ( νn n )n∈N, which are shown always to form a finite subset of (0, 1].
In the final part we introduce a numerical method for calculating the eigenvalues of the standard Laplacian based on a discrete graph approximation and von Below’s theorem.
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Fundação para a Ciência e a Tecnologia
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OE
Número da atribuição
PD/BD/128412/2017