Neumann Laplacian Research Articles

Neumann Laplacian in a Perturbed Domain

We consider a domain with a small compact set of zero Lebesgue measure removed. Our main result concerns the spectrum of the Neumann Laplacian defined on such domain. We prove that the spectrum of the Laplacian converges in the Hausdorff distance sense to the spectrum of the Laplacian defined on the unperturbed domain.

Discrete Laplacian in a half‐space with a periodic surface potential I: Resolvent expansions, scattering matrix, and wave operators

AbstractWe present a detailed study of the scattering system given by the Neumann Laplacian on the discrete half‐space perturbed by a periodic potential at the boundary. We derive asymptotic resolvent expansions at thresholds and eigenvalues, we prove the continuity of the scattering matrix, and we establish new formulas for the wave operators. Along the way, our analysis puts into evidence a surprising relation between some properties of the potential, like the parity of its period, and the behaviour of the integral kernel of the wave operators.

On a conjecture of Ashbaugh and Benguria about lower eigenvalues of the Neumann laplacian

In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the free membrane problem on bounded domains in a Euclidean space or a hyperbolic space which strengthens the well-known Szegö–Weinberger inequality and supports a celebrated conjecture of Ashbaugh–Benguria.

Szegö--Weinberger Type Inequalities for Symmetric Domains with Holes

Let $\mu_2(\Omega)$ be the first positive eigenvalue of the Neumann Laplacian in a bounded domain $\Omega\subset\mathbb{R}^N$. It was proved by Szeg\H{o} for $N=2$ and by Weinberger for $N \geq 2$ that among all equimeasurable domains $\mu_2(\Omega)$ attains its global maximum if $\Omega$ is a ball. In the present work, we develop the approach of Weinberger in two directions. Firstly, we refine the Szeg\H{o}-Weinberger result for a class of domains of the form $\Omega_{\text{out}}\setminus\overline{\Omega}_{\text{in}}$ which are either centrally symmetric or symmetric of order $2$ (with respect to every coordinate plane $(x_i,x_j)$) by showing that $\mu_{2}(\Omega_{\text{out}}\setminus\overline{\Omega}_{\text{in}})\leq\mu_2(B_\beta\setminus\overline{B}_\alpha)$, where $B_\alpha, B_\beta$ are balls centered at the origin such that $B_\alpha\subset\Omega_{\text{in}}$ and $|\Omega_{\text{out}}\setminus\overline{\Omega}_{\text{in}}|=|B_\beta\setminus\overline{B}_\alpha|$. Secondly, we provide Szeg\H{o}-Weinberger type inequalities for higher eigenvalues by imposing additional symmetry assumptions on the domain. Namely, if $\Omega_{\text{out}}\setminus\overline{\Omega}_{\text{in}}$ is symmetric of order $4$, then we prove $\mu_{i}(\Omega_{\text{out}}\setminus\overline{\Omega}_{\text{in}})\leq\mu_i(B_\beta\setminus\overline{B}_\alpha)$ for $i=3,\dots,N+2$, where we also allow $\Omega_{\text{in}}$ and $B_\alpha$ to be empty. If $N=2$ and the domain is symmetric of order $8$, then the latter inequality persists for $i=5$. Counterexamples to the obtained inequalities for domains outside of the considered symmetry classes are given. The existence and properties of nonradial domains with required symmetries in higher dimensions are discussed. As an auxiliary result, we obtain the non-radiality of the eigenfunctions associated to $\mu_{N+2}(B_\beta\setminus\overline{B}_\alpha)$.

Spectral Stability of the {overline{partial }}-Neumann Laplacian: Domain Perturbations

We study spectral stability of the {bar{partial }}-Neumann Laplacian on a bounded domain in {mathbb {C}}^n when the underlying domain is perturbed. In particular, we establish upper semi-continuity properties for the variational eigenvalues of the {bar{partial }}-Neumann Laplacian on bounded pseudoconvex domains in {mathbb {C}}^n, lower semi-continuity properties on pseudoconvex domains that satisfy property (P), and quantitative estimates on smooth bounded pseudoconvex domains of finite D’Angelo type in {mathbb {C}}^n.

$${\text {VMO}}$$ Spaces Associated with Neumann Laplacian

In this paper, we establish several different characterizations of the vanishing mean oscillation space associated with Neumann Laplacian $$\Delta _N$$ , written $${\text {VMO}}_{\Delta _N}({\mathbb {R}^n})$$ . We first describe it with the classical $${\text {VMO}}({\mathbb {R}^n})$$ and certain $${\text {VMO}}$$ on the half-spaces. Then we demonstrate that $${\text {VMO}}_{\Delta _N}({\mathbb {R}^n})$$ is actually $${\text {BMO}}_{\Delta _N}({\mathbb {R}^n})$$ -closure of the space of the smooth functions with compact supports. Beyond that, it can be characterized in terms of compact commutators of Riesz transforms and fractional integral operators associated to the Neumann Laplacian. By means of the functional analysis, we also obtain the duality between certain $${\text {VMO}}$$ and the corresponding Hardy spaces on the half-spaces. Finally, we present an useful approximation for $${\text {BMO}}$$ functions on the space of homogeneous type, which can be applied to our argument and otherwhere.

Superharmonic Double-well Systems with Zero-energy Ground States: Relevance for Diffusive Relaxation Scenarios

Relaxation properties (specifically time-rates) of the Smoluchowski diffusion process on a line, in a confining potential $ U(x) \sim x^m$, $m=2n \geq 2$, can be spectrally quantified by means of the affiliated Schr\"{o}dinger semigroup $\exp (-t\hat{H})$, $t\geq 0$. The inferred (dimensionally rescaled) motion generator $\hat{H}= - \Delta + {\cal{V}}(x)$ involves a potential function ${\cal{V}}(x)= ax^{2m-2} - bx^{m-2}$, $a=a(m), b=b(m) >0$, which for $m>2$ has a conspicuous higher degree (superharmonic) double-well form. For each value of $m>2$, $ \hat{H}$ has the zero-energy ground state eigenfunction $\rho _*^{1/2}(x)$, where $\rho _*(x) \sim \exp -[U(x)]$ stands for the Boltzmann equilibrium pdf of the diffusion process. A peculiarity of $\hat{H}$ is that it refers to a family of quasi-exactly solvable Schr\"{o}dinger-type systems, whose spectral data are either residual or analytically unavailable. As well, no numerically assisted procedures have been developed to this end. Except for the ground state zero eigenvalue and incidental trial-error outcomes, lowest positive energy levels (and energy gaps) of $\hat{H}$ are unknown. To overcome this obstacle, we develop a computer-assisted procedure to recover an approximate spectral solution of $\hat{H}$ for $m>2$. This task is accomplished for the relaxation-relevant low part of the spectrum. By admitting larger values of $m$ (up to $m=104$), we examine the spectral "closeness" of $\hat{H}$, $m\gg 2$ on $R$ and the Neumann Laplacian $\Delta _{\cal{N}}$ in the interval $[-1,1]$, known to generate the Brownian motion with two-sided reflection.

Solvability of a critical semilinear problem with the spectral Neumann fractional Laplacian

Sufficient conditions are provided for the existence of a ground state solution for the problem generated by the fractional Sobolev inequality in Ω ∈ C 2 : \Omega \in C^2: ( − Δ ) S p s u ( x ) + u ( x ) = u 2 s ∗ − 1 ( x ) (-\Delta )_{Sp}^s u(x) + u(x) = u^{2^*_s-1}(x) . Here ( − Δ ) S p s (-\Delta )_{Sp}^s stands for the s s th power of the conventional Neumann Laplacian in Ω ⋐ R n \Omega \Subset \mathbb {R}^n , n ≥ 3 n \geq 3 , s ∈ ( 0 , 1 ) s \in (0, 1) , 2 s ∗ = 2 n / ( n − 2 s ) 2^*_s = 2n/(n-2s) . For the local case where s = 1 s = 1 , corresponding results were obtained earlier for the Neumann Laplacian and Neumann p p -Laplacian operators.

Graph‐like spaces approximated by discrete graphs and applications

AbstractWe define a distance between energy forms on a graph‐like metric measure space and on a suitable discrete weighted graph using the concept of quasi‐unitary equivalence. We apply this result to metric graphs, graph‐like manifolds (e.g. a small neighbourhood of an embedded metric graph) or pcf self‐similar fractals as metric measure spaces with energy forms associated with canonical Laplacians, e.g., the Kirchhoff Laplacian on a metric graph resp. the (Neumann) Laplacian on a manifold (with boundary), and express the distance of the associated energy forms in terms of geometric quantities. In particular, we show that there is a sequence of domains converging to a pcf self‐similar fractal such that the corresponding (Neumann) energy forms converge to the fractal energy form. As a consequence, the spectra and suitable functions of the associated Laplacians converge, the latter in operator norm.

A geometric approximation of δ-interactions by Neumann Laplacians

We demonstrate how to approximate one-dimensional Schrödinger operators with δ-interaction by a Neumann Laplacian on a narrow waveguide-like domain. Namely, we consider a domain consisting of a straight strip and a small protuberance with ‘room-and-passage’ geometry. We show that in the limit when the perpendicular size of the strip tends to zero, and the room and the passage are appropriately scaled, the Neumann Laplacian on this domain converges in generalised norm resolvent sense to the above singular Schrödinger operator. Also we prove Hausdorff convergence of the spectra. In both cases estimates on the rate of convergence are derived.

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.

Bounded H∞-calculus for boundary value problems on manifolds with conical singularities

Realizations of differential operators subject to differential boundary conditions on manifolds with conical singularities are shown to have a bounded H ∞ -calculus in appropriate L p -Sobolev spaces provided suitable conditions of parameter-ellipticity are satisfied. Applications concern the Dirichlet and Neumann Laplacian and the porous medium equation.

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.

Lower bounds for the first eigenvalue of the Laplacian with zero magnetic field in planar domains

We study the Laplacian with zero magnetic field acting on complex functions of a planar domain Ω, with magnetic Neumann boundary conditions. If Ω is simply connected then the spectrum reduces to the spectrum of the usual Neumann Laplacian; therefore we focus on multiply connected domains bounded by convex curves and prove lower bounds for its ground state depending on the geometry and the topology of Ω. Besides the area, the perimeter and the diameter, the geometric invariants which play a crucial role in the estimates are the fluxes of the potential one-form around the inner holes and the distance between the boundary components of the domain; more precisely, the ratio between its minimal and maximal width. Then, we give a lower bound for doubly connected domains which is sharp in terms of this ratio, and a general lower bound for domains with an arbitrary number of holes. When the inner holes shrink to points, we obtain as a corollary a lower bound for the first eigenvalue of the so-called Aharonov-Bohm operators with an arbitrary number of poles.

Wildly perturbed manifolds: norm resolvent and spectral convergence

The publication of the important work of Rauch and Taylor [J. Funct. Anal. 18 (1975)] started a hole branch of research on wild perturbations of the Laplace–Beltrami operator. Here, we extend certain results and show norm convergence of the resolvent. We consider a (not necessarily compact) manifold with many small balls removed, the number of balls can increase as the radius is shrinking, the number of balls can also be infinite. If the distance of the balls shrinks less fast than the radius, then we show that the Neumann Laplacian converges to the unperturbed Laplacian, i.e., the obstacles vanish. In the Dirichlet case, we consider two cases here: if the balls are too sparse, the limit operator is again the unperturbed one, while if the balls concentrate at a certain region (they become “solid” there), the limit operator is the Dirichlet Laplacian on the complement of the solid region. Norm resolvent convergence in the limit case of homogenisation is treated by Khrabustovskyi and the second author in another article (see also the references therein). Our work is based on a norm convergence result for operators acting in varying Hilbert spaces described in a book from 2012 by the second author.

Spectral asymptotic and positivity for singular Dirichlet-to-Neumann operators

In the framework of Hilbert spaces we shall give necessary and sufficient conditions to define a Dirichlet-to-Neumann operator via Dirichlet principle. Analyzing singular Dirichlet-to-Neumann operators, we will establish Laurent expansion near singularities as well as Mittag–Leffler expansion for the related quadratic forms. The established results will be exploited to solve definitively the problem of positivity of the related semigroup in the framework of Lebesgue spaces. The obtained results are supported by some examples where we analyze properties of singular Dirichlet-to-Neumann operators related to Neumann and Robin Laplacian on Lipschitz domains. Among other results, we shall demonstrate that regularity of the boundary may affect positivity and derive Mittag-Leffler expansion for the eigenvalues of singular Dirichlet-to-Neumann operators.

Ambarzumyan-type theorem for the impulsive Sturm–Liouville operator

Abstract We prove that if the Neumann eigenvalues of the impulsive Sturm–Liouville operator - D 2 + q {-D^{2}+q} in L 2 ⁢ ( 0 , π ) {L^{2}(0,\pi)} coincide with those of the Neumann Laplacian, then q = 0 {q=0} .

Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows

We will study metric measure spaces (X,mathsf{d},{mathfrak {m}}) beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds mathsf{BE}_1(kappa ,infty )for which we prove the equivalence with sharp gradient estimates,the class of which will be preserved under time changes with arbitrary psi in mathrm {Lip}_b(X), andwhich are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets Ysubset X. In the latter case, the distribution-valued Ricci bound will be given by the signed measure kappa = k,{mathfrak {m}}_Y + ell ,sigma _{partial Y} where k denotes a variable synthetic lower bound for the Ricci curvature of X and ell denotes a lower bound for the “curvature of the boundary” of Y, defined in purely metric terms. We also present a new localization argument which allows us to pass on the RCD property to arbitrary open subsets of RCD spaces. And we introduce new synthetic notions for boundary curvature, second fundamental form, and boundary measure for subsets of RCD spaces.

Isoperimetric inequalities of the fourth order Neumann eigenvalues

In this paper, we obtain some isoperimetric inequalities for the first (n-1) eigenvalues of the fourth order Neumann Laplacian on bounded domains in an n-dimensional Euclidean space. Our result supports strongly the conjecture of Chasman.

Brownian motion in trapping enclosures: steep potential wells, bistable wells and false bistability of induced Feynman–Kac (well) potentials

We investigate signatures of convergence for a sequence of diffusion processes on a line, in conservative force fields stemming from superharmonic potentials U(x) ∼ x m , m = 2n ⩾ 2. This is paralleled by a transformation of each mth diffusion generator L = DΔ + b(x)∇, and likewise the related Fokker–Planck operator L* = DΔ − ∇[b(x) ⋅], into the affiliated Schrödinger one . Upon a proper adjustment of operator domains, the dynamics is set by semigroups exp(tL), exp(tL*) and exp(−tĤ), with t ⩾ 0. The Feynman–Kac integral kernel of exp(−tĤ) is the major building block of the relaxation process transition probability density, from which L and L* actually follow. The spectral ‘closeness’ of the pertinent Ĥ and the Neumann Laplacian in the interval is analyzed for m even and large. As a byproduct of the discussion, we give a detailed description of an analogous affinity, in terms of the m-family of operators Ĥ with a priori chosen , when Ĥ becomes spectrally ‘close’ to the Dirichlet Laplacian for large m. For completness, a somewhat puzzling issue of the absence of negative eigenvalues for Ĥ with a bistable-looking potential has been addressed.