Neumann Eigenvalue Research Articles

Characterization of potential smoothness and the Riesz basis property of the Hill–Schrödinger operator in terms of periodic, antiperiodic and Neumann spectra

The Hill operators Ly=−y″+v(x)y, considered with complex valued π-periodic potentials v and subject to periodic, antiperiodic or Neumann boundary conditions have discrete spectra. For sufficiently large n, close to n2 there are two periodic (if n is even) or antiperiodic (if n is odd) eigenvalues λn−, λn+ and one Neumann eigenvalue νn. We study the geometry of “the spectral triangle” with vertices (λn+, λn−, νn), and show that the rate of decay of triangle size characterizes the potential smoothness. Moreover, it is proved, for v∈Lp([0,π]),p>1, that the set of periodic (antiperiodic) root functions contains a Riesz basis if and only if for even (respectively, odd) nsupλn+≠λn−{|λn+−νn|/|λn+−λn−|}<∞.

Some inequalities involving eigenvalues of the Neumann Laplacian

AbstractThis paper is concerned with the eigenvalues of the Neumann Laplacian on various classes of domains of given measure: simply‐connected Lipschitz planar domains, n‐sided planar polygons and smooth N‐dimensional domains. In each case, we consider some quantities involving low eigenvalues of the Neumann Laplacian for which we obtain new inequalities. Moreover, we sharpen a universal bound derived by M. Ashbaugh and R. Benguria for sum of reciprocal of Neumann eigenvalues. Our investigations make use of some properties of conformal mappings, Bessel functions, symmetric domains or some isoperimetric inequalities for moments of inertia. Copyright © 2013 John Wiley &amp; Sons, Ltd.

A sharp lower bound for the first eigenvalue on Finsler manifolds

In this paper, we give a sharp lower bound for the first (nonzero) Neumann eigenvalue of Finsler-Laplacian in Finsler manifolds in terms of diameter, dimension, weighted Ricci curvature.

A planar convex domain with many isolated “ hot spots” on the boundary

We construct a convex domain such that the second Neumann eigenfunction has an arbitrary number of isolated local maximum points on the boundary. This domain, which is close to a sector, is constructed by combining thin isosceles triangles. Then we study the shape of the second Neumann eigenfunction on isosceles triangles. In particular, we show that if the isosceles triangle is subequilateral, then the second Neumann eigenvalue is simple and the associated eigenfunction has exactly two maximum points which are located at two corners.

Continuous dependence of eigenvalues of $p$-biharmonic problems on $p$

We are concerned with the Dirichlet and Neumann eigenvalue problem for the ordinary quasilinearfourth-order ($p$-biharmonic) equation\begin{eqnarray}(|u''|^{p-2}u'')''=\lambda|u|^{p-2}u, in \quad [0,1], \quad p>1.\end{eqnarray}It is known that the eigenvalues of the Dirichlet and Neumann $p$-biharmonic problem are positiveand nonnegative, respectively, isolated, and form an increasing unbounded sequence. We prove thatthe eigenvalues depend continuously on $p$, and that they interlace with the eigenvalues of theNavier $p$-biharmonic problem.

Optimal upper bounds for the eigenvalue ratios of one-dimensional $p$-Laplacian

We give optimal upper bounds for the Dirichlet and Neumann eigenvalue ratios of the one-dimensional $p$-Laplacian with nonnegative potentials. In case the potential is single-well, the upper bound for the Dirichlet eigenvalue ratios can be further refined.

Weighted isoperimetric inequalities in cones and applications

This paper deals with weighted isoperimetric inequalities relative to cones of RN. We study the structure of measures that admit as isoperimetric sets the intersection of a cone with balls centered at the vertex of the cone. For instance, in case that the cone is the half-space R+N={x∈RN:xN>0} and the measure is factorized, we prove that this phenomenon occurs if and only if the measure has the form dμ=axNkexp(c|x|2)dx, for somea>0, k,c≥0. Our results are then used to obtain isoperimetric estimates for Neumann eigenvalues of a weighted Laplace–Beltrami operator on the sphere, sharp Hardy-type inequalities for functions defined in a quarter space and, finally, via symmetrization arguments, a comparison result for a class of degenerate PDE’s.

Some results on the Schiffer conjecture in [formula omitted

Let Ω be an open, bounded domain in R2 with connected and C∞ boundary, and ω a solution of(0.1)−△ω=μω,(0.2)∂ω∂n|∂Ω=0,(0.3)ω|∂Ω=const≠0 for some μ>0. Denoting by 0=μ1(Ω)<μ2(Ω)⩽⋯ the set of all Neumann eigenvalues for the Laplacian on Ω, we show that 1) if μ<μ8(Ω); or 2) if Ω is strictly convex and centrally symmetric, μ<μ13(Ω), then Ω must be a disk.

An eigenvalue problem for vibrating strings with concave densities

In this paper, we study the order of the Dirichlet and Neumann eigenvalues for the string equation with concave densities. In general, the order of the k -th Dirichlet eigenvalue and ( k + 1 ) -th Neumann eigenvalue is indeterminate. But, if the density function ρ is concave on ( 0 , 1 ) , we show that the second Neumann eigenvalue is greater than or equal to the first Dirichlet eigenvalue; the equality holds if and only if ρ is constant.

Estimates on the modulus of expansion for vector fields solving nonlinear equations

In this article, by extending the method of Andrews and Clutterbuck (2011) [2] we prove a sharp estimate on the expansion modulus of the gradient of the logarithm of the parabolic kernel to the Schrödinger operator with convex potential on a bounded convex domain. The result improves an earlier work of Brascamp–Lieb which asserts the log-concavity of the parabolic kernel. We also give an alternate proof to a corresponding estimate on the first eigenfunction of the Schrödinger operator, obtained firstly by Andrews and Clutterbuck via the study of the asymptotics to a parabolic problem. Our proof is more direct via an elliptic maximum principle. An alternate proof of the fundamental gap theorem of Andrews and Clutterbuck (2011) [2], by considering the quotient of moduli of continuity, is also obtained. Moreover we derive a Neumann eigenvalue comparison result and some other lower estimates on the first Neumann eigenvalue for Laplace operator with a drifting term, including an explicit estimate on a conjecture of P. Li.

Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup

If $\Omega$ is any compact Lipschitz domain, possibly in a Riemannian manifold, with boundary $\Gamma = \partial \Omega$, the Dirichlet-to-Neumann operator $\mathcal{D}_\lambda$ is defined on $L^2(\Gamma)$ for any real $\lambda$. We prove a close relationship between the eigenvalues of $\mathcal{D}_\lambda$ and those of the Robin Laplacian $\Delta_\mu$, i.e. the Laplacian with Robin boundary conditions $\partial_\nu u =\mu u$. This is used to give another proof of the Friedlander inequalities between Neumann and Dirichlet eigenvalues, $\lambda^N_{k+1} \leq \lambda^D_k$, $k \in N$, and to sharpen the inequality to be strict, whenever $\Omega$ is a Lipschitz domain in $R^d$. We give new counterexamples to these inequalities in the general Riemannian setting. Finally, we prove that the semigroup generated by $-\mathcal{D}_\lambda$, for $\lambda$ sufficiently small or negative, is irreducible.

Eigenvalue estimate for a weighted p-Laplacian on compact manifolds with boundary

Let (Mn, g) be a compact Riemannian manifold with convex boundary, let dµ = eh(x)dV (x) be a weighted measure on M, and let Δµ,p be the corresponding weighted p-Laplacian on M. We obtain a lower bound for the first nonzero Neumann eigenvalue of Δµ,p.

Minimization of the zeroth Neumann eigenvalues with integrable potentials

For an integrable potential q on the unit interval, let λ0(q) be the zeroth Neumann eigenvalue of the Sturm–Liouville operator with the potential q. In this paper we will solve the minimization problem L˜1(r)=infqλ0(q), where potentials q have mean value zero and L1 norm r. The final result is L˜1(r)=−r2/4. The approach is a combination of variational method and limiting process, with the help of continuity results of solutions and eigenvalues of linear equations in potentials and in measures with weak topologies. These extremal values can yield optimal estimates on the zeroth Neumann eigenvalues.

Eigenvalues of collapsing domains and drift Laplacians

By introducing a weight function to the Laplace operator, Bakry and Emery defined the to study diffusion processes. Our first main result is that, given a Bakry-Emery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in R-n and Neumann eigenvalues of domains in Rn+1 and a new maximum principle. Using our main result and maximum principle, we are able to generalize all the results in Riemannian geometry based on gradient estimates to Bakry-Emery manifolds.

A Bifurcation-Type Result for Nonlinear Neumann Eigenvalue Problems

We consider a nonlinear Neumann eigenvalue problem driven by the p-Laplacian and with a (p – 1)-sublinear reaction. Using variational methods together with suitable truncation techniques, we prove a bifurcation-type theorem for the eigenvalue problem. Namely, we show that there is a critical parameter value λ* > 0 such that for all λ > λ* the problem has at least two positive solutions, for λ = λ* there is at least one positive solution and for λ ∈ (0, λ*) no positive solutions exist.

Neumann eigenvalue sums on triangles are (mostly) minimal for equilaterals

We prove that among all triangles of given diameter, the equilateral triangle minimizes the sum of the first n eigenvalues of the Neumann Laplacian, when n 3 . The result fails for n = 2 , because the second eigenvalue is known to be minimal for the degenerate acute isosceles triangle (rather than for the equilateral) while the first eigenvalue is 0 for every triangle. We show the third eigenvalue is minimal for the equilateral triangle. Mathematics subject classification (2010): Primary 35P15. Secondary 35J20.

Isoperimetric inequalities for the first Neumann eigenvalue in Gauss space

We provide isoperimetric Szegö–Weinberger type inequalities for the first nontrivial Neumann eigenvalue μ1(Ω) in Gauss space, where Ω is a possibly unbounded domain of RN. Our main result consists in showing that among all sets Ω of RN symmetric about the origin, having prescribed Gaussian measure, μ1(Ω) is maximum if and only if Ω is the Euclidean ball centered at the origin.

Multiple Constant Sign and Nodal Solutions for Nonlinear Neumann Eigenvalue Problems

We consider a nonlinear Neumann eigenvalue problem driven by a possibly nonhomogeneous differential operator which incorporates as a special case the p-Laplacian. We assume that the right-hand side nonlinearity is (p − 1)-superlinear, but need not satisfy the Ambrosetti-Rabinowitz condition or to be monotone. We show that, for all values of the parameter λ in an upper half line, the problem has two positive and two negative solutions. Subsequently, for the case of the p-Laplacian, we also produce a nodal solution. Finally, for the semilinear case we show that the problem has two nodal solutions. Mathematics Subject Classification (2010): 35J25 (primary); 35J80, 58E05 (secondary).

Convergence Rates in L 2 for Elliptic Homogenization Problems

We study rates of convergence of solutions in L 2 and H 1/2 for a family of elliptic systems $${\{\mathcal{L}_\varepsilon\}}$$ with rapidly oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of $${\{\mathcal{L}_\varepsilon\}}$$ . Most of our results, which rely on the recently established uniform estimates for the L 2 Dirichlet and Neumann problems in Kenig and Shen (Math Ann 350:867–917, 2011; Commun Pure Appl Math 64:1–44, 2011) are new even for smooth domains.

Simple Neumann eigenvalues for the Laplace operator in a domain with a small hole

Let \({\mathbb{A}}(\epsilon)\) be the annular domain obtained by removing from a bounded open domain \({\mathbb{I}}^{o}\) of ℝn a small cavity of size ϵ>0. Then we assume that for some natural index l, \(\lambda_{l}[{\mathbb{I}}^{o}]>0\) is a simple Neumann eigenvalue of −Δ in \({\mathbb{I}}^{o}\), and we show that there exists a real valued real analytic function \(\hat{\lambda }_{l}(\cdot,\cdot)\) defined in an open neighborhood of (0,0) in ℝ2 such that the lth Neumann eigenvalue \(\lambda_{l}[{\mathbb{A}}(\epsilon)]\) of −Δ in \({\mathbb{A}}(\epsilon)\) equals \(\hat{\lambda}_{l}(\epsilon,\kappa_{n}\epsilon\log\epsilon)\) and such that \(\hat{\lambda}_{l}(0,0)= \lambda_{l}[{\mathbb{I}}^{o}]\). Here κn=1 if n is even and κn=0 if n is odd. Thus in particular, we show that if n is even \(\lambda_{l}[{\mathbb {A}}(\epsilon)]\) can be expanded into a convergent double series of powers of ϵ and ϵlogϵ and that if n is odd \(\lambda_{l}[{\mathbb{A}}(\epsilon)]\) can be expanded into a convergent series of powers of ϵ. Then related statements have been proved for corresponding eigenfunctions.