Bounds on the multiplicity of eigenvalues for fixed membranes
Thomas Hoffman-Ostenhof, Peter W. Michor, Nikolai Nadirashvili
对于平面中的膜，已知第$k$个本征值的重数不大于$2k-1$。在这里，我们证明它实际上不大于$2k-3$，对于$k\ge 3$。
For a membrane in the plane the multiplicity of the $k$-th eigenvalue is known to be not greater than $2k-1$. Here we prove that it is actually not greater than $2k-3$, for $k\ge 3$.
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Multiplicity of solutions for gradient systems under strong resonance at the first eigenvalue
本文建立了一类在第一特征值处具有强共振的椭圆型方程组解的存在性和多解性。为了描述共振，我们使用一个具有不定权的本征值问题。在所有结果中，我们使用变分方法。
In 1961 G.Polya published a paper about the eigenvalues of vibrating membrane. The "free vibrating membrane"' corresponds to the Neumann-Laplace operator in bounded plane domains. In this paper we obtain estimates for the first eigenvalue of this operator in a large class of domains that we call as conformal regular domains, that includes convex domains, John domains etc... On the base of our estimates we conjecture that the eigenvalues of the Neumann-Laplace operator depend on the hyperbolic metrics of plane domains. We propose a new method for the estimates that is based on weighted Poincaré…
Upper bound on the multiplicity of eigenvalues of the Schödinger-Dirichlet operator in dimension two
该研究针对二维Schrödinger-Dirichlet算子，建立了其特征值重数的上界，并利用Cheng的广义Morse引理完成证明。
A note on the persistence of multiplicity of eigenvalues of fractional Laplacian under perturbations
本文研究分数拉普拉斯算子特征值重数在扰动下的保持性，给出了保持重数的条件，并证明重数2的特征值对应的保持扰动集为余维2的光滑流形。
Multiplicity of nontrivial solutions for elliptic equations with nonsmooth potential and resonance at higher eigenvalues
考虑一类具有非光滑局部\hbox{Lipschitz}势函数（半变分不等式）的半线性椭圆型方程.我们的假设允许在无穷大和零处的双共振（双双共振情况）。我们的方法是基于局部Lipschitz泛函的非光滑临界点理论，并使用一个抽象的多重性结果下的局部链接和扩展的卡斯特罗-Lazer-Thews减少方法的非光滑设置，我们在这里开发使用的工具，从非光滑分析。
Nodal Sets of Laplacian Eigenfunctions with an Eigenvalue of Multiplicity 2
该研究针对重数为2的简并拉普拉斯特征值，分析矩形单边边界形变对结点集的影响，明确了小扰动下结点集的几何性质变化规律，证明一般形变下内部交叉破裂并给出结点集的位置与正则性估计。
Necessary and sufficient condition for existence at resonance for eigenvalues of multiplicity two
本文研究了半线性狄利克雷问题在二重特征值处解存在的必要且充分条件，并应用该结果讨论了半线性热方程所有解的无界性。
In this short note, we consider the elliptic problem $$ λϕ+ Δϕ= η|ϕ|^σϕ,\quad ϕ\big|_{\partial Ω}=0,\quad λ, η\in \mathbb{C}, $$ on a smooth domain $Ω\subset \mathbb{R}^N$, $N\ge 1$. The presence of complex coefficients, motivated by the study of complex Ginzburg-Landau equations, breaks down the variational structure of the equation. We study the existence of nontrivial solutions as bifurcations from the trivial solution. More precisely, we characterize the bifurcation branches starting from eigenvalues of the Dirichlet-Laplacian of arbitrary multiplicity. This allows us to discuss the nature…
Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes
我们考虑以下特征值最优化问题：给定一个有界域$Ω\subset\R^n$和数$α\geq 0$，$A\in [0,|Ω|]$，找到一个区域$A$的子集$D\subsetΩ$，使算子$-Δ+ αχ_D$的第一Dirichlet特征值尽可能小.我们证明了解的存在性，并研究了它们的定性性质。例如，我们表明，对于一些对称域（薄annuli和哑铃与窄柄）的最优解必须具有较少的对称性比$Ω$ ;另一方面，凸$Ω$反射对称性被保存。此外，我们还给出了数值结果并提出了他们提出的一些猜想。
A generalized Nonlinear Fourier Transform (GNFT), which includes eigenvalues of higher multiplicity, is considered for information transmission over fiber optic channels. Numerical algorithms are developed to compute the direct and inverse GNFTs. For closely-spaced eigenvalues, examples suggest that the GNFT is more robust than the NFT to the practical impairments of truncation, discretization, attenuation and noise. Communication using a soliton with one double eigenvalue is numerically demonstrated, and its information rates are compared to solitons with one and two simple eigenvalues.
Multiplicity of Laplacian eigenvalues that can be represented by sum of two squares using number theory
该研究借助数论证明了二维椭圆 PDE 中关于两平方和型 Laplacian 特征值任意指定奇偶重数的猜想，并给出构造公式。
The Free Boundary Problem in the Optimization of Composite Membranes
这是S. Chanillo，D. Grieser，M. Imai，K.仓田和我。大西再次，我们考虑以下特征值最优化问题：给定有界域$Ω\subset\R^n$和数$α\geq 0$，$A\in [0,|Ω|]$，找到区域$A$的子集$D\subsetΩ$，其中算子$-Δ+ αχ_D$的第一Dirichlet特征值尽可能小。本文主要研究一般区域上最优解的自由边界问题。
We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains $Ω\subset \mathbb{R}^N$. By means of topological arguments, we show how symmetries of $Ω$ help to construct subsets of $W_0^{1,p}(Ω)$ with suitably high Krasnosel'ski\uı genus. In particular, if $Ω$ is a ball $B \subset \mathbb{R}^N$, we obtain the following chain of inequalities: $$ λ_2(p;B) \leq \dots \leq λ_{N+1}(p;B) \leq λ_\ominus(p;B). $$ Here $λ_i(p;B)$ are variational eigenvalues of the $p$-Laplaci…
Membranes with thin and heavy inclusions: asymptotics of spectra
本文研究二维振动系统中闭合曲线附近质量密度扰动导致的特征值渐近，发现极限算子非自伴且谱结构复杂，包含无限重数特征值。
Mixed local-nonlocal eigenvalue problems in the Heisenberg group: spectral theory and singular multiplicity
本文研究海森堡群上混合局部-非局部扩散算子的非线性特征值问题，建立谱理论并证明受扰奇异问题的重数结果，首次系统处理可变奇异指数，为亚椭圆算子谱理论开辟新方向。
We study a Dirichlet-type boundary value problem for a pseudo-differential equation driven by the fractional Laplacian, with a non-linear reaction term which is resonant at infinity between two non-principal eigenvalues: for such equation we prove existence of a non-trivial solution. Under further assumptions on the behavior of the reaction at zero, we detect at least three non-trivial solutions (one positive, one negative, and one of undetermined sign). All results are based on the properties of weighted fractional eigenvalues, and on Morse theory.
We study the transmission eigenvalues for the multipoint scatterers of the Bethe-Peierls-Fermi-Zeldovich-Beresin-Faddeev type in dimensions $d=2$ and $d=3$. We show that for these scatterers: 1) each positive energy $E$ is a transmission eigenvalue (in the strong sense) of infinite multiplicity; 2) each complex $E$ is an interior transmission eigenvalue of infinite multiplicity. The case of dimension $d=1$ is also discussed.
Regularity of the minimizers in the composite membrane problem in R^2
本文研究了平面上复合膜问题极小解的正则性（即给定一个区域Ω和一个小于Ω测度的正数A，极小化具有Ω的子集D的特征函数的固定倍数的势的Schrodinger算子的第一Dirichlet特征值，测度为A）.我们证明了对于极小元，D的边界是解析的。
A celebrated result in bifurcation theory is that global connected sets of non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues of odd algebraic multiplicity of the linearized problem when the operators involved are compact. In this paper a simple example is constructed which satisfies the regularity hypotheses of the global bifurcation theorem, and the eigenvalue has algebraic multiplicity one, yet all the path-connected components of the connected sets that bifurcate are singletons. Another example shows that even when the operators are everywhere infinitely differe…
We consider the following eigenvalue optimization in the composite membrane problem with fractional Laplacian: given a bounded domain $Ω\subset \mathbb{R}^n$, $α>0$ and $0<A<|Ω|$, find a subset $D\subset Ω$ of area $A$ such that the first Dirichlet eigenvalue of the operator $(-Δ)^s+αχ_D$ is as small as possible. The solution $D$ is called as an optimal configuration for the data $(Ω,α,A)$. Looking at the well-known extension definition for the fractional Laplacian, in the case $s=1/2$ this is essentially the composite membrane problem for which the mass is concentrated at the boundary as one …
On Arnold's Transversality Conjecture for the Laplace--Beltrami Operator
本文研究连通流形上黎曼度量集合中使得 Laplace–Beltrami 算子具有给定重数特征值的子集的结构，利用 Besson 的几何刻画证明除无穷余维集合外所有度量满足强 Arnold 假设，并得到任意重数特征值对应度量集合的精确余维。
On the exact number of bifurcation branches from a multiple eigenvalue
研究了一类半线性椭圆型方程重数大于1的特征值的局部分支。我们评估的确切数目的分歧分支的非平凡的解决方案，我们计算的Morse指数的解决方案在这些分支。
We obtained estimates for first eigenvalues of the Dirichlet boundary value problem for elliptic operators in divergence form (i.e. for the principal frequency of non-homogeneous membranes) in bounded domains $Ω\subset \mathbb C$ satisfying quasihyperbolic boundary conditions. The suggested method is based on the quasiconformal composition operators on Sobolev spaces and their applications to constant estimates in the corresponding Sobolev-Poincaré inequalities. We also prove a variant of the Rayleigh-Faber-Khran inequality for a special case of these elliptic operators.
On advective nonlocal operators: multiplicity of principal eigenpairs
本文研究带平流的周期异质非局部扩散模型的主本征对多重性，因算子非预解紧无法用经典Krein-Rutman理论，分平流系数常号与非常号情形分析，还讨论其在非线性KPP型方程中的应用。
It is established existence and multiplicity of solutions for strongly nonlinear problems driven by the $Φ$-Laplacian operator on bounded domains. Our main results are stated without the so called $Δ_{2}$ condition at infinity which means that the underlying Orlicz-Sobolev spaces are not reflexive.
A Vibrating Clamped Plate Resembles a Vibrating Membrane at High Frequency
本研究针对夹紧板特征值问题，证明其特征函数可由满足亥姆霍兹方程的振荡分量近似，误差随频率指数衰减，还建立了其边界局部化的上界，表明高频下夹紧板类似振动膜。
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