Robin Laplacian Research Articles

Isoperimetric inequalities for inner parallel curves

We prove weighted isoperimetric inequalities for smooth, bounded, and simply-connected domains. More precisely, we show that the moment of inertia of inner parallel curves for domains with fixed perimeter attains its maximum for a disk. This inequality, which was previously only known for convex domains, allows us to extend an isoperimetric inequality for the magnetic Robin Laplacian to non-convex centrally symmetric domains. Furthermore, we extend our isoperimetric inequality for moments of inertia, which are second moments, to p -th moments for all p smaller than or equal to two. We also show that the disk is a strict local maximiser in the nearly circular, centrally symmetric case for all p strictly less than three, and that the inequality fails for all p strictly bigger than three.

Boundary states of the Robin magnetic Laplacian

This article tackles the spectral analysis of the Robin Laplacian on a smooth bounded two-dimensional domain in the presence of a constant magnetic field. In the semiclassical limit, a uniform description of the spectrum located between the Landau levels is obtained. The corresponding eigenfunctions, called edge states, are exponentially localized near the boundary. By means of a microlocal dimensional reduction, our unifying approach allows on the one hand to derive a very precise Weyl law and a proof of quantum magnetic oscillations for excited states, and on the other hand to refine simultaneously old results about the low-lying eigenvalues in the Robin case and recent ones about edge states in the Dirichlet case.

Multidimensional Borg–Levinson uniqueness and stability results for the Robin Laplacian with unbounded potential

This article deals with the uniqueness and stability issues in the inverse problem of determining the unbounded potential of the Schrödinger operator in a bounded domain of \mathbb{R}^{n} , n\geq 3 , endowed with Robin boundary condition, from knowledge of its boundary spectral data. These data are defined by the pairs formed by the eigenvalues and either partial or full Dirichlet measurement of the eigenfunctions on the boundary of the domain.

Comparison results for solutions of Poisson equations with Robin boundary on complete Riemannian manifolds

In this paper, by using Schwarz rearrangement and isoperimetric inequalities, we prove comparison results for the solutions of Poisson equations on complete Riemannian manifolds with [Formula: see text], [Formula: see text] or [Formula: see text], which extends the results in [A. Alvino, C. Nitsch and C. Trombetti, A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions, Comm. Pure Appl. Math. 76(3) (2023) 585–603]. Furthermore, as applications of our comparison results, we obtain the Saint-Venant inequality and Bossel–Daners inequality for Robin Laplacian.

Magnetic perturbations of the Robin Laplacian in the strong coupling limit

This paper is devoted to the asymptotic analysis of the eigenvalues of the Laplace operator with a strong magnetic field and Robin boundary condition on a smooth planar domain and with a negative boundary parameter. We study the singular limit when the Robin parameter tends to infinity, which is equivalent to a semi-classical limit involving a small positive semi-classical parameter. The main result is a comparison between the spectrum of the Robin Laplacian with an effective operator defined on the boundary of the domain via the Born–Oppenheimer approximation. More precisely, the low-lying eigenvalue of the Robin Laplacian is approximated by those of the effective operator. When the curvature has a unique non-degenerate maximum, we estimate the spectral gap and find that the magnetic field does not contribute to the three-term expansion of the eigenvalues. In the case of the disc domains, the eigenvalue asymptotics displays the contribution of the magnetic field explicitly.

Non-monotonicity of the first eigenvalue for the 3D magnetic Robin Laplacian

Previous works provided several counterexamples to monotonicity of the lowest eigenvalue for the magnetic Laplacian in the two-dimensional case. However, the three-dimensional case is less studied. We use the results obtained by Helffer, Kachmar, and Raymond to provide one of the first counterexamples in 3D. Considering the magnetic Robin Laplacian on the unit ball with a constant magnetic field, we show the non-monotonicity of the lowest eigenvalue asymptotics when the Robin parameter tends to +infty .

Semilinear elliptic equations on rough domains

The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the usual arguments based on maximum principles do not apply. A key result is an application of Kato's inequality to prove a comparison theorem for eigenfunctions that only requires interior regularity and avoids the use of the Hopf boundary maximum principle. We demonstrate the theory on an abstract degenerate logistic equation by proving the existence, uniqueness and stability of non-trivial positive solutions. Examples of operators include the Dirichlet Laplacian on arbitrary bounded domains, a simplified construction of the Robin Laplacian on arbitrary domains with boundary of finite measure and general elliptic operators in divergence form.

Spectral isoperimetric inequalities for Robin Laplacians on 2-manifolds and unbounded cones

We consider the problem of geometric optimization of the lowest eigenvalue for the Laplacian on a compact, simply-connected two-dimensional manifold with boundary subject to an attractive Robin boundary condition. We prove that in the sub-class of manifolds with the Gauss curvature bounded from above by a constant K_\circ \ge 0 and under the constraint of fixed perimeter, the geodesic disk of constant curvature K_\circ maximizes the lowest Robin eigenvalue. In the same geometric setting, it is proved that the spectral isoperimetric inequality holds for the lowest eigenvalue of the Dirichlet-to-Neumann operator. Finally, we adapt our methods to Robin Laplacians acting on unbounded three-dimensional cones to show that, under a constraint of fixed perimeter of the cross-section, the lowest Robin eigenvalue is maximized by the circular cone.

Operator estimates for homogenization of the Robin Laplacian in a perforated domain

Let ε>0 be a small parameter. We consider the domain Ωε:=Ω∖Dε, where Ω is an open domain in Rn, and Dε is a family of small balls of the radius dε=o(ε) distributed periodically with period ε. Let Δε be the Laplace operator in Ωε subject to the Robin condition ∂u∂n+γεu=0 with γε≥0 on the boundary of the holes and the Dirichlet condition on the exterior boundary. Kaizu (1985, 1989) and Brillard (1988) have shown that, under appropriate assumptions on dε and γε, the operator Δε converges in the strong resolvent sense to the sum of the Dirichlet Laplacian in Ω and a constant potential. We improve this result deriving estimates on the rate of convergence in terms of L2→L2 and L2→H1 operator norms. As a byproduct we establish the estimate on the distance between the spectra of the associated operators.

Faber-Krahn inequalities for the Robin Laplacian on bounded domain in Riemannian manifolds

In this paper, we obtain the Faber-Krahn inequality for the first eigenvalue of the Robin Laplacian on bounded domain in Riemannian manifolds whose Ricci curvature satisfies Ricg≥(n−1). The Faber-Krahn inequality also holds for the Robin Laplacian on bounded domain in hyperbolic space Hn.

Correction to: On the Isoperimetric Inequality for the Magnetic Robin Laplacian with Negative Boundary Parameter

Correction to: On the Isoperimetric Inequality for the Magnetic Robin Laplacian with Negative Boundary Parameter

Resolvent expansions for self‐adjoint operators via boundary triplets

In this paper, we develop certain aspects of perturbation theory for self-adjoint operators subject to small variations of their domains. We use the abstract theory of boundary triplets to quantify such perturbations and give the second-order asymptotic analysis for resolvents, spectral projections, and discrete eigenvalues of the corresponding self-adjoint operators. In particular, we derive explicit formulas for the first variation and the Hessian of the eigenvalue curves bifurcating from a discrete eigenvalue of an unperturbed operator. An application is given to a matrix valued Robin Laplacian and more general Robin-type self-adjoint extensions.

Spectral optimization for Robin Laplacian on domains admitting parallel coordinates

AbstractIn this paper we deal with spectral optimization for the Robin Laplacian on a family of planar domains admitting parallel coordinates, namely a fixed‐width strip built over a smooth closed curve and the exterior of a convex set with a smooth boundary. We show that if the curve length is kept fixed, the first eigenvalue referring to the fixed‐width strip is for any value of the Robin parameter maximized by a circular annulus. Furthermore, we prove that the second eigenvalue in the exterior of a convex domain Ω corresponding to a negative Robin parameter does not exceed the analogous quantity for the exterior of a disk whose boundary has a curvature larger than or equal to the maximum of that for .

On the Isoperimetric Inequality for the Magnetic Robin Laplacian with Negative Boundary Parameter

We consider the magnetic Robin Laplacian with a negative boundary parameter on a bounded, planar $$C^2$$ -smooth domain. The respective magnetic field is homogeneous. Among a certain class of domains, we prove that the disk maximises the ground-state energy under the fixed perimeter constraint provided that the magnetic field is of moderate strength. This class of domains includes, in particular, all domains that are contained upon translations in the disk of the same perimeter and all convex centrally symmetric domains.

Semi‐classical edge states for the Robin Laplacian

Motivated by the study of high-energy Steklov eigenfunctions, we examine the semi-classical Robin Laplacian. In the two-dimensional situation, we determine an effective operator describing the asymptotic distribution of the negative eigenvalues, and we prove that the corresponding eigenfunctions decay away from the boundary, for all dimensions.

On the eigenvalues of the Robin Laplacian with a complex parameter

We study the spectrum of the Robin Laplacian with a complex Robin parameter \(\alpha \) on a bounded Lipschitz domain \(\Omega \). We start by establishing a number of properties of the corresponding operator, such as generation properties, analytic dependence of the eigenvalues and eigenspaces on \(\alpha \in {\mathbb {C}}\), and basis properties of the eigenfunctions. Our focus, however, is on bounds and asymptotics for the eigenvalues as functions of \(\alpha \): we start by providing estimates on the numerical range of the associated operator, which lead to new eigenvalue bounds even in the case \(\alpha \in {\mathbb {R}}\). For the asymptotics of the eigenvalues as \(\alpha \rightarrow \infty \) in \({\mathbb {C}}\), in place of the min–max characterisation of the eigenvalues and Dirichlet–Neumann bracketing techniques commonly used in the real case, we exploit the duality between the eigenvalues of the Robin Laplacian and the eigenvalues of the Dirichlet-to-Neumann map. We use this to show that along every analytic curve of eigenvalues, the Robin eigenvalues either diverge absolutely in \({\mathbb {C}}\) or converge to the Dirichlet spectrum, as well as to classify all possible points of accumulation of Robin eigenvalues for large \(\alpha \). We also give a comprehensive treatment of the special cases where \(\Omega \) is an interval, a hyperrectangle or a ball. This leads to the conjecture that on a general smooth domain in dimension \(d\ge 2\) all eigenvalues converge to the Dirichlet spectrum if \({\mathrm{Re}}\,\alpha \) remains bounded from below as \(\alpha \rightarrow \infty \), while if \({\mathrm{Re}}\,\alpha \rightarrow -\infty \), then there is a family of divergent eigenvalue curves, each of which behaves asymptotically like \(-\alpha ^2\).

Effective operators for changing sign Robin Laplacian in thin two- and three-dimensional curved waveguides

Effective operators for changing sign Robin Laplacian in thin two- and three-dimensional curved waveguides

Generalized nonlocal Robin Laplacian on arbitrary domains

In this paper, we prove that it is always possible to define a realization of the Laplacian \(\Delta _{\kappa ,\theta }\) on \(L^2(\Omega )\) subject to nonlocal Robin boundary conditions with general jump measures on arbitrary open subsets of \({\mathbb {R}}^N\). This is made possible by using a capacity approach to define an admissible pair of measures \((\kappa ,\theta )\) that allows the associated form \({\mathcal {E}}_{\kappa ,\theta }\) to be closable. The nonlocal Robin Laplacian \(\Delta _{\kappa ,\theta }\) generates a sub-Markovian \(C_0\)-semigroup on \(L^2(\Omega )\) which is not dominated by the Neumann Laplacian semigroup unless the jump measure \(\theta \) vanishes.

Disjointness-preserving operators and isospectral Laplacians

The most commonly considered counterexamples to Kac’s famous question “can one hear the shape of a drum?” – i.e., does isospectrality of two Laplacians on domains imply that the domains are congruent? – consist of pairs of domains composed of copies of isometric building blocks arranged in different ways, such that the unitary operator intertwining the Laplacians acts as a sum of overlapping “local” isometries mapping the copies to each other. We prove and explore a complementary positive statement: if an operator intertwining two appropriate realisations of the Laplacian on a pair of domains preserves disjoint supports, then under additional assumptions on it generally far weaker than unitarity, the domains are congruent. We show this in particular for the Dirichlet, Neumann, and Robin Laplacians on spaces of continuous functions and on L^2 -spaces.

New eigenvalue estimates involving Bessel functions

Given a compact Riemannian manifold (Mn, g) with boundary ∂M, we give an estimate for the quotient R ∂M f dµg R M f dµg , where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [39], we provide a differential inequality for f which, under some curvature assumptions, can be interpreted in terms of Bessel functions. As an application of our main result, a new inequality is given for Dirichlet and Robin Laplacian. Also, a new estimate is established for the eigenvalues of the Dirac operator that involves a positive root of Bessel function besides the scalar curvature. Indepen[1]dently, we extend the Robin Laplacian on functions to differential forms. We prove that this natural extension defines a self-adjoint and elliptic operator whose spectrum is discrete and consists of positive real eigenvalues. In particular, we characterize its first eigenvalue and provide a lower bound of it in terms of Bessel functions.