Daniel Daners, James B. Kennedy
We prove that every eigenvalue of a Robin problem with boundary parameter $α$ on a sufficiently smooth domain behaves asymptotically like $-α^2$ as $α\to \infty$. This generalises an existing result for the first eigenvalue.
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We study some properties of Laplacian eigenvalues with negative Robin boundary conditions. We will show some monotonicity properties on annuli of the first eigenvalue by means of shape optimization techniques.
In this paper we study the main properties of the first eigenvalue and its eigenfunctions of a class of highly nonlinear elliptic operators in a bounded Lipschitz domain, assuming a Robin boundary condition. Moreover, we prove a Faber-Krahn inequality.
Upper and lower bounds for the first Robin eigenvalue of nonlinear elliptic operators
本文研究了非线性椭圆算子第一Robin特征值的估计问题，证明了涉及$p$-laplacian算子第一Robin特征值的不等式，特别是利用第一Dirichlet特征值给出了该特征值的上界。
Geometrical bounds for the torsion and the first eigenvalue of the Laplacian with Robin boundary condition
本文研究了Robin边界条件下拉普拉斯算子扭力和第一特征值与几何量的关系，证明了Robin扭力的上界及相关形状函数的最优值由滑动域实现。
In this paper, we prove an upper bound for the first Robin eigenvalue of the $p$-Laplacian with a positive boundary parameter and a quantitative version of the reverse Faber-Krahn type inequality for the first Robin eigenvalue of the $p$-Laplacian with negative boundary parameter, among convex sets with prescribed perimeter. The proofs are based on a comparison argument obtained by means of inner sets, introduced by Payne, Weimberger and Polya.
We prove the existence of an optimal partition for the multiphase shape optimization problem which consists in minimizing the sum of the first Robin Laplacian eigenvalue of $k$ mutually disjoint {\it open} sets which have a $\mathcal H ^ {d-1}$-countably rectifiable boundary and are contained into a given box $D$ in $R^d$
The considered Robin problem can formally be seen as a small perturbation of a Dirichlet problem. However, due to the sign of the impedance value, its associated eigenvalues converge point-wise to $-\infty$ as the perturbation goes to zero. We prove that in this case, Dirichlet eigenpairs are the only accumulation points of the Robin eigenpairs with normalized eigenvectors. We then provide a criteria to select accumulating sequences of eigenvalues and eigenvectors and exhibit their full asymptotic with respect to the small parameter.
The paper is devoted to the study of some properties of the first eigenvalue of the anisotropic $p$-Laplace operator with Robin boundary condition involving a function $β$ which in general is not constant. In particular we obtain sharp lower bounds in terms of the measure of the domain and we prove a monotonicity property of the eigenvalue with respect the set inclusion.
On a shape derivative formula for the Robin $p$-Laplace eigenvalue
本文推导了Robin $p$-Laplace算子第一特征值的形状导数公式，并利用该公式证明在边界参数较大时，光滑区域上的第一特征值关于区域包含关系具有单调性。
On the second eigenvalue of the infinity Laplacian with Robin boundary conditions
研究Robin边界条件下p-拉普拉斯算子第二特征值在p趋于无穷时的极限，证明其收敛于无穷拉普拉斯算子的第二特征值。
Relations between principal eigenvalue and torsional rigidity with Robin boundary conditions
本文研究了Robin边界条件下主特征值与扭屈刚度的关系，确定了临界指数并展示了其与狄利克雷情况的差异。
We consider a superlinear perturbation of the eigenvalue problem for the Robin Laplacian plus an indefinite and unbounded potential. Using variational tools and critical groups, we show that when $λ$ is close to a nonprincipal eigenvalue, then the problem has seven nontrivial solutions. We provide sign information for six of them.
We prove two bounds for the first Robin eigenvalue of the Finsler Laplacian with negative boundary parameter in the planar case. In the constant area problem, we show that the Wulff shape is the maximizer only for values which are close to 0 of the boundary parameter and, in the fixed perimeter case, that the Wulff shape maximizes the first eigenvalue for all values of the parameter.
Simplicity of eigenvalues for elliptic problems with mixed Steklov-Robin boundary condition
本文研究了混合Steklov-Robin边界条件下的椭圆问题，证明了特征值的简性，通过领域扰动技术和算子横截性分析实现。
In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools, truncation techniques and comparison methods. The existence of the obtained solutions depends on the first eigenvalues of the Robin and Steklov eigenvalue problems for the $p$-Laplacian.
This document consists of two papers, both submitted, and supplementary material. The submitted papers are here given as Parts I and II. Part I establishes results, used in Part II, 'on functions and inverses, both positive, decreasing and convex'. Part II uses results from Part I to extablish 'inequalities for the fundamental Robin eigenvalue for the Laplacian on N-dimensional boxes'
The asymptotic behavior of the first Robin eigenvalue with negative parameter as $p$ goes to $+\infty$
本文研究了带有负边界参数的第一 $p$-Laplacian Robin 特征值在 $p$ 趋向于正无穷时的渐近行为，并证明其特征函数的极限是 infinity Laplacian 特征值问题的粘性解。
Let Ωbe a bounded connected, open set of \R^n with Lipschitz boundary. Let F be a suitable norm in \R^n and let Δ_F u be the so-colled Finsler Laplacian. In this paper we prove two inequalities for the first eigenvalue of Δ_F with Robin boundary conditions involving a positive function β. As a consequence of our result we obtain the asymptotic behavior of this eigenvalue when βis a positive constant which goes to zero.
A sharp bound for the first Robin-Dirichlet eigenvalue
针对双连通区域上外边界Robin、内边界Dirichlet条件的拉普拉斯第一特征值问题，证明了固定测度、外周长与内第(n-1)次均质积分的区域中球壳使该特征值取最大值，给出精确上界。
Inverse Iteration for the Laplace Eigenvalue Problem with Robin and Mixed Boundary Conditions
本文将反迭代法应用于带Robin和混合Dirichlet-Neumann边界条件的Laplace特征值问题，证明了迭代序列向主特征函数及Rayleigh商向主特征值的收敛性，并针对最优绝缘模型提出了相关迭代方法及部分结果。
On the optimization of the Robin eigenvalues in some classes of polygons
该研究针对带Robin边界条件的拉普拉斯算子特征值问题，在边数有上界的广义多边形类中，分别在周长或体积约束下解决了$\beta>0$时的形状极小化与$\beta<0$时的形状极大化问题。
Sharp stability on the second Robin eigenvalue with negative boundary parameters
本文研究了负边界参数下第二Robin特征值的稳定性，通过构造近球形域族证明了等周不等式中不对称度指数的尖锐性。
Minimization of the $k$ -th eigenvalue of the Robin-Laplacian with perimeter constraint
本文针对带周长约束的 Robin-Laplacian 第 $k$ 个特征值最小化问题，推广至含内部裂缝的有限周长集合，证明极小值存在且等于正则域下确界。
In this paper we study a Steklov-Robin eigenvalue problem for the Laplacian in annular domains. More precisely, we consider $Ω=Ω_0 \setminus \overline{B}_{r}$, where $B_{r}$ is the ball centered at the origin with radius $r>0$ and $Ω_0\subset\mathbb{R}^n$, $n\geq 2$, is an open, bounded set with Lipschitz boundary, such that $\overline{B}_{r}\subset Ω_0$. We impose a Steklov condition on the outer boundary and a Robin condition involving a positive $L^{\infty}$-function $β(x)$ on the inner boundary. Then, we study the first eigenvalue $σ_β(Ω)$ and its main properties. In particular, we investi…
The eigenvalue problem for the p-Laplace operator with Robin boundary condition is considered in this paper. A Faber-Krahn type inequality is proved. More precisely, it is shown that amongst all the domains of fixed volume, the ball has the smallest first eigenvalue.
We consider the problem of minimising the $k$th eigenvalue, $k \geq 2$, of the ($p$-)Laplacian with Robin boundary conditions with respect to all domains in $\mathbb{R}^N$ of given volume $M$. When $k=2$, we prove that the second eigenvalue of the $p$-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For $p=2$ and $k \geq 3$, we prove that in many cases a minimiser cannot be independent of the value of the constant $α$ in the boundary condition, or equivalently of the volume $M$. We obtain …
Principal eigenvalues and asymptotic behavior for the weighted $p$-Laplacian with Robin boundary conditions on exterior domains
本文研究了外域上带Robin边界条件的加权p-拉普拉斯主特征值及其渐进行为，推导了连接近边界和远场区域的统一梯度估计。
A Faber-Krahn inequality for the Laplacian with drift under Robin boundary condition
本研究证明了大β参数下Robin边界条件带漂移拉普拉斯算子的Faber-Krahn不等式，并论证了满足L^∞约束、使对应算子主本征值取极值的漂移的存在唯一性。
We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear perturbation. For the first case we show that for $λ<\widehatλ_{1}$ ($\widehatλ_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $λ\geq\widehatλ_{1}$ there are no positive solutions. In the superlinear case, for $λ<\widehatλ_{1}$ we have at least two positive solutions and for…
An isoperimetric inequality for the second Robin eigenvalue of the Weighted Laplacian
本文证明在特定条件下，球体是最大化加权拉普拉斯算子第二罗宾特征值的最优形状。
The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the third eigenvalue of a disjoint union of two disks, provided the Robin parameter lies in a certain range and is scaled in each case by the length of the boundary. Equality is achieved when the domain degenerates suitably to the two disks.
Upper bounds for the ratio of the first two eigenvalues of the Robin Laplacian
本文证明了Robin拉普拉斯算子前两个特征值比值的最大值在所有维度和边界参数下均由圆盘取得，并证明该比值随边界参数递减。
By applying Mountain Pass Lemma, Ekeland's and Ricceri's variational principle, Fountain Theorem, we prove the existence and multiplicity of solutions for the following Robin problem \begin{equation*} \left\{ \begin{array}{cc} -\text{div}\left( a(x)\left\vert \nabla u\right\vert ^{p(x)-2}\nabla u\right) =λb(x)\left\vert u\right\vert ^{q(x)-2}u, & x\in Ω\\ a(x)\left\vert \nabla u\right\vert ^{p(x)-2}\frac{\partial u}{\partial \upsilon }+β(x)\left\vert u\right\vert ^{p(x)-2}u=0, & x\in \partial Ω, \end{array} \right. \end{equation*} under some appropriate conditions in the space $W_{a,b}^{1,p(.)…
Inverse Robin Spectral Problem for the p-Laplace Operator
本文研究了p-拉普拉斯算子的逆罗宾谱问题，通过渐近分析和线性化方法确定未知罗宾系数，并给出了稳定性估计。
We study the Robin eigenvalue problem for the Laplace-Beltrami operator on Riemannian manifolds. Our first result is a comparison theorem for the second Robin eigenvalue on geodesic balls in manifolds whose sectional curvatures are bounded from above. Our second result asserts that geodesic balls in nonpositively curved space forms maximize the second Robin eigenvalue among bounded domains of the same volume.
We consider a nonlinear Robin problems driven by the $p$-Laplacian plus an indefinite potential. The reaction is resonant with respect to a variational eigenvalue. For the principal eigenvalue we assume strong resonance. Using variational tools and critical groups we prove existence and multiplicity theorems.
In this paper, we prove two results for the Robin eigenvalue problem. One is an upper bound for the ratio of the first two eigenvalues which can be used to recover the PPW conjecture proved by M.S.Ashbaugh and R.D.Benguria, the other is a reverse Holder inequality for the first eigenfunction which is a natural generalization of Chiti's reverse Holder inequality for the first eigenfunction of Dirichlet Laplacian.
Geometric Optimization of the First Robin Eigenvalue in Exterior Domains
该研究将二维下球使离散第一Robin特征值最大化的结论推广到高维，证明近球形区域中球是局部极大值点，但一般并非全局极大值点。
We consider the first eigenvalue $λ_1(Ω,σ)$ of the Laplacian with Robin boundary conditions on a compact Riemannian manifold $Ω$ with smooth boundary, $σ\in\bf R$ being the Robin boundary parameter. When $σ>0$ we give a positive, sharp lower bound of $λ_1(Ω,σ)$ in terms of an associated one-dimensional problem depending on the geometry through a lower bound of the Ricci curvature of $Ω$, a lower bound of the mean curvature of $\partialΩ$ and the inradius. When the boundary parameter is negative, the lower bound becomes an upper bound. In particular, explicit bounds for mean-convex Euclidean do…
We prove that the optimal cluster problem for the sum of the first Robin eigenvalue of the Laplacian, in the limit of a large number of convex cells, is asymptotically solved by (the Cheeger sets of) the honeycomb of regular hexagons. The same result is established for the Robin torsional rigidity.
Weighted Robin eigenvalue problems and nonlinear elliptic equations with general growth in the gradient
本文研究了带权Robin特征值问题与具有一般增长梯度的非线性椭圆方程的存在性，通过引入带权奇异特征值问题，证明了在特定条件下解的存在性。
The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $Θ$ which give rise to self-adjoint Laplacians $-Δ_{Θ, Ω}$ in $L^2(Ω; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $Ω\subset\bbR^n$, $n\in\bbN$, $n\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $Ω$, following an approach introduced by Filonov for this type of problems.
We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two-sided estimates for this term in a variety of situations.
On a bounded Lipschitz domain we consider two selfadjoint operator realizations of the same second order elliptic differential expression subject to Robin boundary conditions, where the coefficients in the boundary conditions are functions. We prove that inequality between these functions on the boundary implies strict inequality between the eigenvalues of the two operators, provided that the inequality of the functions in the boundary conditions is strict on an arbitrarily small nonempty, open set.
In this paper we prove that the ball maximizes the first eigenvalue of the Robin Laplacian operator with negative boundary parameter, among all convex sets of \mathbb{R}^n with prescribed perimeter. The key of the proof is a dearrangement procedure of the first eigenfunction of the ball on the level sets of the distance function to the boundary of the convex set, which controls the boundary and the volume energies of the Rayleigh quotient.
On the solutions to $p$-Poisson equation with Robin boundary conditions when $p$ goes to $+\infty$
研究当 $p$ 趋于无穷大时带有 Robin 边界条件的 $p$-Laplacian 特征值及相关特征函数的极限行为，证明该极限是 $\infty$-Laplacian 特征值问题的粘性解。
Among all $C^{\infty}$ bounded domains with equal volume, we show that the second eigenvalue of the Robin plate is uniquely maximized by an open ball, so long as the Robin parameter lies within a particular range of negative values. Our methodology combines recent techniques introduced by Freitas and Laugesen to study the second eigenvalue of the Robin membrane problem and techniques employed by Chasman to study the free plate problem. In particular, we choose eigenfunctions of the ball as trial functions in the Rayleigh quotient for a general domain; such eigenfunctions are comprised of ultra…
In this paper, we prove a quantitative spectral inequality for the second Robin eigenvalue in non-compact rank-1 symmetric spaces. In particular, this shows that for bounded domains in non-compact rank-1 symmetric spaces, the geodesic ball maximises the second Robin eigenvalue among domains of the same volume, with negative Robin parameter in the regime connecting the first nontrivial Neumann and Steklov eigenvalues. This result generalises the work of Freitas and Laugesen in the Euclidean setting [FL21] as well as our previous work in the hyperbolic space [LWW20].
A stability result for the first Robin-Neumann eigenvalue: A double perturbation approach
该研究针对带Robin外边界、Neumann内边界的拉普拉斯算子第一特征值问题，引入新型混合不对称性，建立了等周不等式的定量版本，得到首个内外边界同时扰动的稳定性结果。
We study the low-lying eigenvalues of the semiclassical Robin Laplacian in a smooth planar domain symmetric with respect to an axis. In the case when the curvature of the boundary of the domain attains its maximum at exactly two points away from the axis of symmetry, we establish an explicit asymptotic formula for the splitting of the first two eigenvalues. This is a rigorous derivation of the semiclassical tunneling effect induced by the domain's geometry. Our approach is close to the Born-Oppenheimer one and yields, as a byproduct, a Weyl formula of independent interest.
Laplacian eigenvalues with a large negative Robin parameter on a part of the boundary
本研究针对部分边界带强吸引Robin条件、其余为Neumann条件的光滑平面区域拉普拉斯特征值问题，借助区间有效算子刻画单特征值渐近行为，并针对典型几何得到基于边界曲率的更精确渐近结果。
On asymptotics of Robin eigenvalues in the Dirichlet limit
本文研究了 Robin 参数发散时 Laplacian 算子 Robin 特征值向 Dirichlet 特征值收敛的渐近行为，通过引入新的扭转刚度概念，证明了特征值的一阶展开及特征函数的收敛速率，且谱估计显式且尖锐。
Laplacian eigenvalues for large negative Robin parameters on domains with outward peaks
本文研究具有向外尖峰的域上，大负Robin参数下拉普拉斯特征值的渐近行为，推广了横截面为球的情形，结果反映尖峰尖锐度与横截面形状。
The second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $α/L(Ω)$, and $α$ lies between $-2π$ and $2π$. Corollaries include Szegő's sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock's inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter. The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization o…
In this paper we prove a Faber-Krahn type inequality for the first eigenvalue of the Hermite operator with Robin boundary condition. We prove that the optimal set is an half-space and we also address the equality case in such inequality.
We prove a quantitative Faber-Krahn inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions. The asymmetry term involves the square power of the Fraenkel asymmetry, multiplied by a constant depending on the Robin parameter, the dimension of the space and the measure of the set.
We consider the first Robin eigenvalue $ł_p(M,\a)$ for the $p$-Laplacian on a compact Riemannian manifold $M$ with nonempty smooth boundary, with $\a \in \R$ being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of Cheng type for $ł_p(M,\a)$. Secondly, when $\a>0$ we establish sharp lower bound of $ł_p(M,\a)$ in terms of dimension, inradius, Ricci curvature lower bound and boundary mean curvature lower bound, via comparison with an associated one-dimensional eigenvalue problem. The lower bound becomes an upper bound when $\a<0$. Our results cover corresponding comparison …
Homogenisation for the Robin eigenvalue problem on manifolds and flexibility of optimal Schrödinger potentials
本文利用齐次化构造 Robin 问题逼近闭流形及带边紧致流形上的 Schrödinger 谱，并证明最优势可由在对偶 Sobolev 空间中远离它的光滑势序列逼近。
In this paper, we are interested in the analysis of a well-known free boundary/shape optimization problem motivated by some issues arising in population dynamics. The question is to determine optimal spatial arrangements of favorable and unfavorable regions for a species to survive. The mathematical formulation of the model leads to an indefinite weight linear eigenvalueproblem in a fixed box $Ω$ and we consider the general case of Robin boundary conditions on $\partialΩ$. It is well known that it suffices to consider {\it bang-bang} weights taking two values of different signs, that can be pa…
The first two eigenvalues of the Robin Laplacian are investigated along with their gap and ratio. Conjectures by various authors for arbitrary domains are supported here by new results for rectangular boxes. Results for rectangular domains include that: the square minimizes the first eigenvalue among rectangles under area normalization, when the Robin parameter $α\in \mathbb{R}$ is scaled by perimeter; that the square maximizes the second eigenvalue for a sharp range of $α$-values; that the line segment minimizes the Robin spectral gap under diameter normalization for each $α\in \mathbb{R}$; a…
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