I. Birindelli, S. Patrizi
In this paper we study the asymptotic behavior of the principal eigenvalues associated to the Pucci operator in bounded domain $Ω$ with Neumann/Robin boundary condition i.e. $\partial_n u=αu$ when $α$ tends to infinity. This study requires Lipschitz estimates up to the boundary that are interesting in their own rights.
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Nonlinear Neumann eigenvalues in outward cuspidal domains with weighted measure
该研究针对带权测度外尖点区域的非线性Neumann特征值问题，通过Sobolev空间复合算子建立到带权Lebesgue空间的嵌入，证明了谱问题可解性并给出特征值估计。
We prove the existence of a principal eigenvalue associated to the $\infty$-Laplacian plus lower order terms and the Neumann boundary condition in a bounded smooth domain. As an application we get uniqueness and existence results for the Neumann problem and a decay estimate for viscosity solutions of the Neumann evolution problem.
We study the variation of the Neumann eigenvalues of the $p$-Laplace operator under quasiconformal perturbations of space domains. This study allows to obtain lower estimates of the Neumann eigenvalues in fractal type domains. The suggested approach is based on the geometric theory of composition operators in connections with the quasiconformal mapping theory.
Here we prove an isoperimetric inequality for the harmonic mean of the first $N-1$ non-trivial Neumann eigenvalues of the Laplace-Beltrami operator for domains contained in a hemisphere of $\mathbb{S}^N$.
In this paper we show the existence of two principal eigenvalues associated to general non-convex fully nonlinear elliptic operators with Neumann boundary conditions in a bounded $C^2$ domain. We study these objects and we establish some of their basic properties. Finally, Lipschitz regularity, uniqueness and existence results for the solution of the Neumann problem are given.
In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of R N with prescribed measure m attains its maximum on the union of two disjoint balls of measure m 2. As a consequence, the P{ó}lya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.
We deal with the sharp asymptotic behaviour of eigenvalues of elliptic operators with varying mixed Dirichlet-Neumann boundary conditions. In case of simple eigenvalues, we compute explicitly the constant appearing in front of the expansion's leading term. This allows inferring some remarkable consequences for Aharonov-Bohm eigenvalues when the singular part of the operator has two coalescing poles.
In this paper we discuss applications of the geometric theory of composition operators on Sobolev spaces to the spectral theory of non-linear elliptic operators. The lower estimates of the first non-trivial Neumann eigenvalues of the $p$-Laplace operator in cusp domains $Ω\subset\mathbb R^n$, $n\geq 2$, are given.
In this paper we prove a sharp lower bound for the first nontrivial Neumann eigenvalue $μ_1(Ω)$ for the $p$-Laplace operator in a Lipschitz, bounded domain $Ω$ in $\R^n$. Our estimate does not require any convexity assumption on $Ω$ and it involves the best isoperimetric constant relative to $Ω$.
The first nontrivial eigenfunction of the Neumann eigenvalue problem for the $p$-Laplacian, suitable normalized, converges as $p$ goes to $\infty$ to a viscosity solution of an eigenvalue problem for the $\infty$-Laplacian. We show among other things that the limit of the eigenvalue, at least for convex sets, is in fact the first nonzero eigenvalue of the limiting problem. We then derive a number of consequences, which are nonlinear analogues of well-known inequalities for the linear (2-)Laplacian.
In this paper we apply estimates of the norms of Sobolev extension operators to the spectral estimates of of the first nontrivial Neumann eigenvalue of the Laplace operator in non-convex extension domains. As a consequence we obtain a connection between resonant frequencies of free membranes and the smallest-circle problem (initially proposed by J.~J.~Sylvester in 1857).
This paper deals with the Neumann eigenvalue problem for the Hermite operator defined in a convex, possibly unbounded, planar domain $Ω$, having one axis of symmetry passing through the origin. We prove a sharp lower bound for the first eigenvalue $μ_1^{odd}(Ω)$ with an associated eigenfunction odd with respect to the axis of symmetry. Such an estimate involves the first eigenvalue of the corresponding one-dimensional problem. As an immediate consequence, in the class of domains for which $μ_1(Ω)=μ_1^{odd}(Ω)$, we get an explicit lower bound for the difference between $μ(Ω)$ and the first Neum…
In this work we study the asymptotic behavior of the first non-zero Neumann $p-$fractional eigenvalue $λ_1(s,p)$ as $s\to 1^-$ and as $p\to\infty.$ We show that there exists a constant $\mathcal{K}$ such that $\mathcal{K}(1-s)λ_1(s,p)$ goes to the first non-zero Neumann eigenvalue of the $p-$Laplacian. While in the limit case $p\to \infty,$ we prove that $λ_1(1,s)^{1/p}$ goes to an eigenvalue of the Hölder $\infty-$Laplacian.
We develop some properties of the $p-$Neumann derivative for the fractional $p-$Laplacian in bounded domains with general $p>1$. In particular, we prove the existence of a diverging sequence of eigenvalues and we introduce the evolution problem associated to such operators, studying the basic properties of solutions. Finally, we study a nonlinear problem with source in absence of the Ambrosetti-Rabinowitz condition.
In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the free membrane problem on bounded domains of a Euclidean space or a hyperbolic space which strengthens the well-known Szegö-Weinberger inequality and supports strongly an important conjecture of Ashbaugh-Benguria.
We study spectral properties of divergence form elliptic operators $-\textrm{div} [A(z) \nabla f(z)]$ with the Neumann boundary condition in planar domains (including some fractal type domains), that satisfy to the quasihyperbolic boundary conditions. Our method is based on an interplay between quasiconformal mappings, elliptic operators and composition operators on Sobolev spaces.
In this article we obtain estimates of Neumann eigenvalues of $p$-Laplace operators in a large class of space domains satisfying quasihyperbolic boundary conditions. The suggested method is based on composition operators generated by quasiconformal mappings and their applications to Sobolev-Poincaré-inequalities. By using a sharp version of the inverse Hölder inequality we refine our estimates for quasi-balls, that is, images of balls under quasiconformal mappings of the whole space.
Sharp bounds and geometric properties of the first non trivial Steklov Neumann Eigenvalue
本文研究双连通域上的混合Steklov-Neumann特征值问题，证明当双连通域为同心球时，第一非零特征值最大，并推导了双连通星型域和球形孔域的特征值界及渐进行为，还研究了相应特征函数的节点域数量。
Magnetic Neumann problems with Aharonov-Bohm potentials: boundary asymptotics of eigenvalues and splitting phenomena
研究了具有Aharonov-Bohm势的磁Neumann问题，通过规范变换分析边界渐进行为，并揭示了极点接近边界时特征值的分裂现象。
On the Neumann $(p,q)$-eigenvalue problem in Hölder singular domains
本文研究Hölder奇异区域中的Neumann $(p,q)$-特征值问题，证明其可解性、相关变分极小元的存在性，并给出特征函数正则性与特征值估计。
Sobolev $(p,q)$-extension operators and Neumann eigenvalues
本文研究了Sobolev空间上的$(p,q)$-扩展算子，通过构造向外尖突域的扩展算子并估计其范数，证明了$p$-Laplace算子在向外尖突域中的非线性Neumann特征值的估计。
Eigenvalues of the Neumann Laplacian with density and sharp Sobolev-Orlicz embeddings
本文针对特殊平面区域和权重估计加权Poincaré常数，进而证明带密度Neumann拉普拉斯算子第一非零特征值的下界，并指出该特征值可通过改变质量密度任意增大。
The anisotropic $\infty$-Laplacian eigenvalue problem with Neumann boundary conditions
本研究借助粘性解分析带诺伊曼边界的各向异性$p$-拉普拉斯当$p\rightarrow\infty$时的极限问题，证明特征值与特征函数的几何性质，验证Szegö-Weinberger型不等式的成立。
This article investigates how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain in the case when the domains involved are of class $C^1$. We consider the Laplacian and use results developed previously for the corresponding Lipschitz case. In contrast with the Lipschitz case however, in the $C^1$-case we derive an asymptotic formula for the eigenvalues when the domains are of class $C^1$. Moreover, as an application we consider the case of a $C^1$-perturbation when the reference domain is of class $C^{1,α}$.
Estimates on the Neumann and Steklov principal eigenvalues of collapsing domains
本文研究了 $\mathbb{R}^2$ 中塌缩凸域的 Neumann 和 Steklov 主特征值关系，部分证明了两者比值的估计猜想，指出变薄三角形最大化该比值，变薄矩形在具有对称性的凸变薄集中最小化该比值。
In this work we are concerned with the multiplicity of the eigenvalues of the Neumann Laplacian in regions of Rn which are invariant under the natural action of a compact subgroup G of O(n). We give a partial positive answer (in the Neumann case) to a conjecture of V. Arnold [1] on the transversality of the transformation given by the Dirichlet integral to the stratification in the space of quadratic forms according to the multiplicities of the eigenvalues. We show, for some classes of subgroups of O(n) that, generically in the set of G-invariant, C2-regions, the action is irreducible in each …
We prove that the second nontrivial Neumann eigenvalue of the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^n \subseteq \mathbb{R}^{n+1}$ is maximized by the union of two disjoint, equal, geodesic balls among all subsets of $\mathbb{S}^n$ of prescribed volume. In fact, the result holds in a stronger version, involving the harmonic mean of the eigenvalues of order $2$ to $n$, and extends to densities. A (surprising) consequence occurs on the maximality of a geodesic ball for the first nontrivial eigenvalue under the volume constraint: the hemisphere inclusion condition of the Ashbaug…
We study the positive principal eigenvalue of a weighted problem associated with the Neumann spectral fractional Laplacian. This analysis is related to the investigation of the survival threshold in population dynamics. Our main result concerns the optimization of such threshold with respect to the fractional order $s\in(0,1]$, the case $s=1$ corresponding to the standard Neumann Laplacian: when the habitat is not too fragmented, the principal positive eigenvalue can not have local minima for $0<s<1$. As a consequence, the best strategy for survival is either following the diffusion with $s=1$…
Estimates of variational eigenvalues on metric measure spaces
本文在倍增度量测度空间上证明变分Neumann (p,q)-特征值极小元的存在性，并据此给出Neumann特征值的估计。
We first investigate spectral properties of the Neumann-Poincaré (NP) operator for the Lamé system of elasto-statics. We show that the elasto-static NP operator can be symmetrized in the same way as that for Laplace operator. We then show that even if elasto-static NP operator is not compact even on smooth domains, its spectrum consists of eigenvalues which accumulates to two numbers determined by Lamé constants. We then derive explicitly eigenvalues and eigenfunctions on disks and ellipses. We then investigate resonance occurring at eigenvalues and anomalous localized resonance at accumulatio…
Principal eigenvalue for some elliptic operators with large drift: Neumann boundary conditions
本文研究二维带漂移线性椭圆算子的主特征值，精细刻画了漂移率趋于无穷时其渐近行为由对应常微分方程组ω极限集的连通分支决定的规律，并讨论了退化情形。
Three non-zero solutions of a Neumann eigenvalue problems involving the fractional p-Laplacian
本文研究含分数阶p-Laplacian的非局部Neumann特征值问题，证明存在正特征值开区间，使问题在该区间内至少有三个非零解。
In this note we analyze how perturbations of a ball $\mathfrak{B}_r \subset \mathbb{R}^n$ behaves in terms of their first (non-trivial) Neumann and Dirichlet $\infty-$eigenvalues when a volume constraint $\\mathscr{L}^n(Ω) = \mathscr{L}^n(\mathfrak{B}_r)$ is imposed. Our main result states that $Ω$ is uniformly close to a ball when it has first Neumann and Dirichlet eigenvalues close to the ones for the ball of the same volume $\mathfrak{B}_r$. In fact, we show that, if $$ |λ_{1,\infty}^D(Ω) - λ_{1,\infty}^D(\mathfrak{B}_r)| = δ_1 \quad \text{and} \quad |λ_{1,\infty}^N(Ω) - λ_{1,\infty}^N(\mat…
In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of biharmonic Stekloff eigenvalues with Neumann boundary condition in a bounded domain of an $n$-dimensional Riemannian manifold.
We deal with eigenvalue problems for the Laplacian with varying mixed boundary conditions, consisting in homogeneous Neumann conditions on a vanishing portion of the boundary and Dirichlet conditions on the complement. By the study of an Almgren type frequency function, we derive upper and lower bounds of the eigenvalue variation and sharp estimates in the case of a strictly star-shaped Neumann region.
In this paper we present a preliminary study on the Dirichlet-to-Neumann operator with respect to a second order elliptic operator with measurable coefficients, including first order terms, namely, the operator on $L^2(\partialΩ)$ given by $φ\mapsto \partial_νu$ where $u$ is a weak solution of \begin{equation} \left\{ \begin{aligned} -{\rm div}\, (a\nabla u) +b\cdot \nabla u -{\rm div}\, (cu)+du & =λu \ \ \text{on}\ Ω,\\ u|_{\partialΩ} & =φ. \end{aligned} \right. \end{equation} Under suitable assumptions on the matrix-valued function $a$, on the vector fields $b$ and $c$, and on the function $…
We give new estimates on the lower bounds for the first closed or Neumann eigenvalue for a compact manifold with positive Ricci curvature in terms of the diameter and the lower bound of Ricci curvature. The results improve the previous estimates.
Eigenvalues of nonlinear $(p,q)$-fractional Laplace operators under nonlocal Neumann conditions
研究非局部 $(p,q)$-分数阶 Laplace 算子在齐次非局部 Neumann 条件下的特征值问题，完整刻画其谱集并证明特征函数全局有界。
This paper is devoted to a comparison between the normalized first (non-trivial) Neumann eigenvalue $|Ω| μ_1(Ω)$ for a Lipschitz open set $Ω$ in the plane, and the normalized first (non-trivial) Steklov eigenvalue $P(Ω) σ_1(Ω)$. More precisely, we study the ratio $F(Ω):=|Ω| μ_1(Ω)/P(Ω) σ_1(Ω)$. We prove that this ratio can take arbitrarily small or large values if we do not put any restriction on the class of sets $Ω$. Then we restrict ourselves to the class of plane convex domains for which we get explicit bounds. We also study the case of thin convex domains for which we give more precise bo…
We study the eigenvalue problem for the Neumann-Laplace operator in conformal regular planar domains $Ω\subset\mathbb{C}$. Conformal regular domains support the Poincaré inequality and this allows us to estimate the variation of the eigenvalues of the Neumann Laplacian upon domain perturbation via energy type integrals. Boundaries of such domains can have any Hausdorff dimension between one and two.
We use the notion of first and second inner variations as a bridge allowing one to pass to the limit of first and second Gateaux variations for the Allen-Cahn, Cahn-Hilliard and Ohta-Kawasaki energies. Under suitable assumptions, this allows us to show that stability passes to the sharp interface limit, including boundary terms, by considering non-compactly supported velocity and acceleration fields in our variations. This complements the results of Tonegawa, and Tonegawa and Wickramasekera, where interior stability is shown to pass to the limit. As a further application, we prove an asymptoti…
A note on the distribution of Neumann eigenvalues of the Laplacian on a Euclidean convex domain
本文研究了欧几里得凸域上拉普拉斯算子Neumann特征值的分布，建立了两个通用不等式。
The first terms of the small volume asymptotic expansion for the splitting of Neumann boundary condition Laplacian eigenvalues due to a grounded inclusion of size ε are derived. An explicit formula to compute the first term from the eigenvalues and eigenfunctions of the unperturbed domain, the inclusion size and position is given. As a consequence, when an eigenvalue of double multiplicity splits in two distinct eigenvalues, one decays like O(1/log(ε)), the other like O(ε^2).
Inequalities between Dirichlet and Neumann eigenvalues on Carnot groups
本文证明Carnot群开集上次Laplacian算子的第j个Dirichlet特征值大于第(j+1)个Neumann特征值，推广了Euclidean与Heisenberg情形的早期结果。
We consider the Dirichlet-to-Neumann operator (DNO) on nearly-circular and nearly-spherical domains in two and three dimensions, respectively. Treating such domains as perturbations of the ball, we prove the analyticity of the DNO with respect to the domain perturbation parameter. Consequently, the Steklov eigenvalues are also shown to be analytic in the domain perturbation parameter. To obtain these results, we use the strategy of Nicholls and Nigam (2004); we transform the equation on the perturbed domain to a ball and geometrically bound the Neumann expansion of the transformed Dirichlet-to…
We give an overview of results on shape optimization for low eigenvalues of the Laplacian on bounded planar domains with Neumann and Steklov boundary conditions. These results share a common feature: they are proved using methods of complex analysis. In particular, we present modernized proofs of the classical inequalities due to Szego and Weinstock for the first nonzero Neumann and Steklov eigenvalues. We also extend the inequality for the second nonzero Neumann eigenvalue, obtained recently by Nadirashvili and the authors, to non-homogeneous membranes with log-subharmonic densities. In the h…
Let $Ω$ be a bounded domain in $\mathbb R^2$ with smooth boundary $\partialΩ$, and let $ω_h$ be the set of points in $Ω$ whose distance from the boundary is smaller than $h$. We prove that the eigenvalues of the biharmonic operator on $ω_h$ with Neumann boundary conditions converge to the eigenvalues of a limiting problem in the form of system of differential equations on $\partialΩ$.
An isoperimetric inequality for lower order Neumann eigenvalues in Gauss space
本文在高斯空间中证明了关于低阶Neumann特征值调和平均的等周不等式，并推广了Szegö-Weinberger型不等式。
We provide isoperimetric Szegö-Weinberger type inequalities for the first nontrivial Neumann eigenvalue $μ_{1}(Ω)$ in Gauss space, where $Ω$ is a possibly unbounded domain of $\mathbb{R}^{N}$. Our main result consists in showing that among all sets of $\mathbb{R}^{N}$ symmetric about the origin, having prescribed Gaussian measure, $μ_{1}(Ω)$ is maximum if and only if $Ω$ is the euclidean ball centered at the origin.
This paper is devoted to the study of Lyapunov-type inequality for Neumann boundary conditions at higher eigenvalues. Our main result is derived from a detailed analysis about the number and distribution of zeros of nontrivial solutions and their first derivatives, together with the use of suitable minimization problems. This method of proof allows to obtain new information on Lyapunov constants. For instance, we prove that as in the classical result by Lyapunov, the best constant is not attained. Additionally, we exploit the relation between Neumann boundary conditions and disfocality to prov…
The paper deals with the Neumann spectral problem for a singularly perturbed second order elliptic operator with bounded lower order terms. The main goal is to provide a refined description of the limit behaviour of the principal eigenvalue and eigenfunction. Using the logarithmic transformation we reduce the studied problem to additive eigenvalue problem for a singularly perturbed Hamilton-Jacobi equation. Then assuming that the Aubry set of the Hamiltonian consists of a finite number of points or limit cycles situated in the domain or on its boundary, we find the limit of the eigenvalue and …
In this paper, we establish a priori estimates for a class of fully nonlinear equations with Neumann boundary conditions. By the continuity method, we have obtained the existence theorem for the Neumann problem.
The first Neumann eigenvalue and the width
本文针对有界凸平面区域，证明了第一Neumann特征值可由直径和宽度给出的精确下界，揭示了特征值与区域几何尺度之间的定量关系。
In this paper we study the Neumann problem for a type of fully nonlinear second order elliptic partial differential equations on domains in $\mathbb{C}^{n}$ without any curvature assumptions on the domain.
Two new universal inequalities for Neumann eigenvalues of the Laplacian on a planar convex domain
本文提出两个新通用不等式，用于平面凸域中拉普拉斯算子的Neumann特征值，扩展了相关理论的应用范围。
A note on domain monotonicity for the Neumann eigenvalues of the Laplacian
本文证明了凸区域及其子区域上拉普拉斯算子诺伊曼特征值的区域单调性变体，并基于该方法给出了凸区域诺伊曼特征值的上界。
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 $n \geq 3$. 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.
We show that the third eigenvalue of the Neumann Laplacian in hyperbolic space is maximal for the disjoint union of two geodesic balls, among domains of given volume. This extends a recent result by Bucur and Henrot in Euclidean space, while providing a new proof of a key step in their argument
We consider the Neumann-Poincar'e (double layer potential) operator in 3D elasicity on a smooth closed surface. Its essential spectrum consists of 3 points. We find the asymptotics of sequences of eigenvalues converging to these three pounts. They are expressed via the eigenvalue distribution for a matrix eigenvalue problem depending on the points of the cotangent bundle of the surface. For the two-sided asymptotics for the distribution of the the union of sequences of eigenvalues converging to the point of the essential spectrum from both sides, an expression is found via the Euler characteri…
For a given bounded domain $Ω\subset {\Bbb R}^n$ with $C^1$-smooth boundary, we prove the Pólya conjecture for the Neumann eigenvalues. In other words, we prove that \begin{eqnarray*} μ_{k+1}\le \frac{(2π)^2k^{2/n}}{(ω_n \cdot \mbox{vol}\, (Ω))^{2/n}} \quad \;\; \mbox{for all} \;\; k=0,1,2,3,\cdots,\end{eqnarray*} where $μ_k$ is the $k$-th Neumann eigenvalue of the Laplacian for $Ω$.
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