Recent zbMATH articles in MSC 35P15https://zbmath.org/atom/cc/35P152021-06-15T18:09:00+00:00WerkzeugShape derivative for some eigenvalue functionals in elasticity theory.https://zbmath.org/1460.490322021-06-15T18:09:00+00:00"Caubet, Fabien"https://zbmath.org/authors/?q=ai:caubet.fabien"Dambrine, Marc"https://zbmath.org/authors/?q=ai:dambrine.marc"Mahadevan, Rajesh"https://zbmath.org/authors/?q=ai:mahadevan.rajeshBlaschke-Santaló diagram for volume, perimeter, and first Dirichlet eigenvalue.https://zbmath.org/1460.490332021-06-15T18:09:00+00:00"Ftouhi, Ilias"https://zbmath.org/authors/?q=ai:ftouhi.ilias"Lamboley, Jimmy"https://zbmath.org/authors/?q=ai:lamboley.jimmyOn a Pólya functional for rhombi, isosceles triangles, and thinning convex sets.https://zbmath.org/1460.520082021-06-15T18:09:00+00:00"van den Berg, Michiel"https://zbmath.org/authors/?q=ai:van-den-berg.michiel"Ferone, Vincenzo"https://zbmath.org/authors/?q=ai:ferone.vincenzo"Nitsch, Carlo"https://zbmath.org/authors/?q=ai:nitsch.carlo"Trombetti, Cristina"https://zbmath.org/authors/?q=ai:trombetti.cristinaThe authors state three results. The first one focuses on providing explicitly an upper bound of the product of the first eigenvalue of the Laplacian operator by the norm of the torsion function for an open bounded convex set. The second (resp. third) concentrates on furnishing accurately a lower and an upper bound of Pólya function for an isosceles triangle (resp. a rhombus).
Reviewer: Mohammed El Aïdi (Bogotá)Interface asymptotics of eigenspace Wigner distributions for the harmonic oscillator.https://zbmath.org/1460.352412021-06-15T18:09:00+00:00"Hanin, Boris"https://zbmath.org/authors/?q=ai:hanin.boris-l"Zelditch, Steve"https://zbmath.org/authors/?q=ai:zelditch.steveSummary: Eigenspaces of the quantum isotropic Harmonic Oscillator \(\widehat{H}_{\hslash} := -\frac{\hslash^2}{2} \Delta + \frac12 \|x\|^2\) on \(\mathbb R^d\) have extremally high multiplicites and the eigenspace projections \(\Pi_{\hslash,E_N(\hslash)}\) have special asymptotic properties. This article gives a detailed study of their Wigner distributions \(W_{\hslash,E_N(\hslash)}(x, \xi)\). Heuristically, if \(E_N(\hslash) = E\), \(W_{\hslash,E_N(\hslash)}(x, \xi)\) is the ``quantization'' of the energy surface \(\Sigma_E\), and should be like the delta-function \(\delta_{\Sigma_E}\) on \(\Sigma_E\); rigorously, \(W_{\hslash,E_N(\hslash)}(x,\xi)\) tends in a weak\(^*\) sense to \(\delta_{\Sigma_E}\). But its pointwise asymptotics and scaling asymptotics have more structure. The main results give Bessel asymptotics of \(W_{\hslash,E_N(\hslash)}(x,\xi)\) in the interior \(H(x,\xi) < E\) of \(\Sigma_E\); \textit{interface Airy scaling asymptotics} in tubes of radius \(\hslash^{2/3}\) around \(\Sigma_E\), with \((x,\xi)\) either in the interior or exterior of the energy ball; and exponential decay rates in the exterior of the energy surface.An isoperimetric inequality for the harmonic mean of the Steklov eigenvalues in hyperbolic space.https://zbmath.org/1460.352422021-06-15T18:09:00+00:00"Verma, Sheela"https://zbmath.org/authors/?q=ai:verma.sheelaFor any bounded and smooth domain \(\Omega\) of the standard hyperbolic space \(\mathbb H^n\), an inequality for harmonic mean of the first \(n\) positive eigenvalues \(\mu_1(\Omega),...,\mu_n(\Omega)\) of the Steklov problem on \(\Omega\) is proved, namely \[ \sum_{i=1}^n\frac{1}{\mu_i(\Omega)}\geq\sum_{i=1}^n\frac{1}{\mu_i(B_R)}, \] where \(B_R\) is the ball in \(\mathbb H^n\) with the same volume of \(\Omega\). Equality holds if and only if \(\Omega\) is a ball. The proof relies on the construction of a suitable set of test functions for the Rayleigh quotient in terms of normal coordinate functions.
Reviewer: Luigi Provenzano (Padova)Preconditioners for multilevel Toeplitz linear systems from steady-state and evolutionary advection-diffusion equations.https://zbmath.org/1460.651342021-06-15T18:09:00+00:00"Lin, Xue-lei"https://zbmath.org/authors/?q=ai:lin.xuelei"Ng, Micheal K."https://zbmath.org/authors/?q=ai:ng.micheal-k"Wathen, Andy"https://zbmath.org/authors/?q=ai:wathen.andrew-jSummary: In this paper, we study preconditioners for multilevel Toeplitz linear systems arising from discretization of steady-state and evolutionary advection-diffusion equations, in which upwind scheme and central difference scheme are employed to discretize first-order and second-order terms, respectively. For the steady-state case, the preconditioner is constructed by replacing each of the discrete advection terms with a square root of the negative of discrete Laplacian matrix and the so constructed preconditioner is diagonalizable by a sine transform. Due to its diagonalizability, the preconditioner can be applied in a two-sided way. We prove that the GMRES solver for the preconditioned linear system has a linear convergence rate independent of discretization step-sizes. The sum of the time discretization and the steady-state preconditioner constitutes the evolutionary preconditioner. A fast implementation is proposed for the evolutionary preconditioner. Moreover, for the evolutionary case, we prove that the modulus of the eigenvalues of the preconditioned matrix is lower and upper bounded by positive constants independent of discretization step-sizes. We test the proposed preconditioners with several Krylov subspace solvers on some advection-dominated advection-diffusion problems and compare their performance with other preconditioners to show its efficiency.Robin spectral rigidity of the ellipse.https://zbmath.org/1460.352432021-06-15T18:09:00+00:00"Vig, Amir"https://zbmath.org/authors/?q=ai:vig.amirSummary: In this paper, we investigate \(C^1\) isospectral deformations of the ellipse with Robin boundary conditions, allowing both the Robin function and domain to deform simultaneously. We prove that if the deformations preserve the reflectional symmetries of the ellipse, then the first variation of both the domain and Robin function must vanish. If the deformation is in fact smooth, reparametrizing allows us to show that the first variation actually vanishes to infinite order. In particular, there exist no such analytic isospectral deformations. The key ingredients are a version of Hadamard's variational formula for variable Robin boundary conditions and an oscillatory integral representation of the wave trace variation which uses action angle coordinates for the billiard map. For the latter, we in fact construct an explicit parametrix for the wave propagator in the interior, microlocally near geodesic loops.Sharp estimates for the first \(p\)-Laplacian eigenvalue and for the \(p\)-torsional rigidity on convex sets with holes.https://zbmath.org/1460.351792021-06-15T18:09:00+00:00"Paoli, Gloria"https://zbmath.org/authors/?q=ai:paoli.gloria"Piscitelli, Gianpaolo"https://zbmath.org/authors/?q=ai:piscitelli.gianpaolo"Trani, Leonardo"https://zbmath.org/authors/?q=ai:trani.leonardoSummary: We study, in dimension \(n \geq 2\), the eigenvalue problem and the torsional rigidity for the \(p\)-Laplacian on convex sets with holes, with external Robin boundary conditions and internal Neumann boundary conditions. We prove that the annulus maximizes the first eigenvalue and minimizes the torsional rigidity when the measure and the external perimeter are fixed.Geometric control of the Robin Laplacian eigenvalues: the case of negative boundary parameter.https://zbmath.org/1460.490312021-06-15T18:09:00+00:00"Bucur, Dorin"https://zbmath.org/authors/?q=ai:bucur.dorin"Cito, Simone"https://zbmath.org/authors/?q=ai:cito.simoneSummary: This paper is motivated by the study of the existence of optimal domains maximizing the \(k\)th Robin Laplacian eigenvalue among sets of prescribed measure, in the case of a negative boundary parameter. We answer positively to this question and prove an existence result in the class of measurable sets and for quite general spectral functionals. The key tools of our analysis rely on tight isodiametric and isoperimetric geometric controls of the eigenvalues. In two dimensions of the space, under simply connectedness assumptions, further qualitative properties are obtained on the optimal sets.Quantum Hamiltonians with weak random abstract perturbation. II: Localization in the expanded spectrum.https://zbmath.org/1460.352402021-06-15T18:09:00+00:00"Borisov, Denis"https://zbmath.org/authors/?q=ai:borisov.denis-i"Täufer, Matthias"https://zbmath.org/authors/?q=ai:taufer.matthias"Veselić, Ivan"https://zbmath.org/authors/?q=ai:veselic.ivanSummary: We consider multi-dimensional Schrödinger operators with a weak random perturbation distributed in the cells of some periodic lattice. In every cell the perturbation is described by the translate of a fixed abstract operator depending on a random variable. The random variables, indexed by the lattice, are assumed to be independent and identically distributed according to an absolutely continuous probability density. A small global coupling constant tunes the strength of the perturbation. We treat analogous random Hamiltonians defined on multi-dimensional layers, as well. For such models we determine the location of the almost sure spectrum and its dependence on the global coupling constant. In this paper we concentrate on the case that the spectrum expands when the perturbation is switched on. Furthermore, we derive a Wegner estimate and an initial length scale estimate, which together with Combes-Thomas estimate allow to invoke the multi-scale analysis proof of localization. We specify an energy region, including the bottom of the almost sure spectrum, which exhibits spectral and dynamical localization. Due to our treatment of general, abstract perturbations our results apply at once to many interesting examples both known and new.
For Part I, see [the first author et al., Ann. Henri Poincaré 17, No. 9, 2341--2377 (2016; Zbl 1348.82039)].Lower estimates on eigenvalues of quantum graphs.https://zbmath.org/1460.340362021-06-15T18:09:00+00:00"Mugnolo, Delio"https://zbmath.org/authors/?q=ai:mugnolo.delio"Plümer, Marvin"https://zbmath.org/authors/?q=ai:plumer.marvinSummary: A method for estimating the spectral gap along with higher eigenvalues of quantum graphs has been introduced by \textit{O. Amini} and \textit{D. Cohen-Steiner} [Comment. Math. Helv. 93, No. 1, 203--223 (2018; Zbl 1383.05192)] recently: it is based on a new transference principle between discrete and continuous models of a graph. We elaborate on it by developing a more general transference principle and by proposing alternative ways of applying it. To illustrate our findings, we present several spectral estimates on planar metric graphs that are oftentimes sharper than those obtained by isoperimetric inequalities and further previously known methods.A stability result for the Steklov Laplacian eigenvalue problem with a spherical obstacle.https://zbmath.org/1460.350862021-06-15T18:09:00+00:00"Paoli, Gloria"https://zbmath.org/authors/?q=ai:paoli.gloria"Piscitelli, Gianpaolo"https://zbmath.org/authors/?q=ai:piscitelli.gianpaolo"Sannipoli, Rossanno"https://zbmath.org/authors/?q=ai:sannipoli.rossannoSummary: In this paper we study the first Steklov-Laplacian eigenvalue with an internal fixed spherical obstacle. We prove that the spherical shell locally maximizes the first eigenvalue among nearly spherical sets when both the internal ball and the volume are fixed.