Krzysztof Burdzy
We present a new method for locating the nodal line of the second eigenfunction for the Neumann problem in a planar domain. The technique is based on the `mirror coupling' of reflected Brownian motions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We investigate the measure of nodal sets for Robin and Neumann eigenfunctions in the domain and on the boundary of the domain. A polynomial upper bound for the interior nodal sets is obtained for Robin eigenfunctions in the smooth domain. For the analytic domain, the sharp upper bounds of the interior nodal sets was shown for Robin eigenfunctions. More importantly, we obtain the sharp upper bounds for the boundary nodal sets of Neumann eigenfunctions with new quantitative global Carleman estimates. Furthermore, the sharp doubling inequality and vanishing order of Robin eigenfunctions on the bo…
We prove estimates for eigenfunctions on a manifold equipped with a smooth metric. We use these estimates in order estimate the size of their nodal sets.
We consider the lower bound of nodal sets of Steklov eigenfunctions on smooth Riemannian manifolds with boundary--the eigenfunctions of the Dirichlet-to-Neumann map. Let $N_λ$ be its nodal set. Assume that zero is a regular value of Steklov eigenfunctions. We show that $$H^{n-1}(N_λ)\geq Cλ^{\frac{3-n}{2}}$$ for some positive constant $C$ depending only on the manifold.
Let ϕ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set {ϕ=0}.
In this paper, we derive the sharp lower and upper bounds of nodal lengths of Laplacian eigenfunctions in the disc. Furthermore, we observe a geometric property of the eigenfunctions whose nodal curves maximize the nodal length.
In this paper we will prove the nodal line $N$ of the second eigenfunction of the Laplacian over some simply connected concave domain $Ω$ in $\mathbb{R}^2$ must intersect the boundary $\partialΩ$ at exactly two points.
The aim of this article is to provide a simple and unified way to obtain the sharp upper bounds of nodal sets of eigenfunctions for different types of eigenvalue problems on real analytic domains. The examples include biharmonic Steklov eigenvalue problems, buckling eigenvalue problems and champed-plate eigenvalue problems. The geometric measure of nodal sets are derived from doubling inequalities and growth estimates for eigenfunctions. It is done through analytic estimates of Morrey-Nirenberg and Carleman estimates.
We prove lower bounds for the Hausdorff measure of nodal sets of eigenfunctions.
We introduce the class of quasiconvex Lipschitz domains, which covers both $C^1$ and convex domains, to the study of boundary unique continuation for elliptic operators. In particular, we prove the upper bound of the size of nodal sets for Dirichlet eigenfunctions of general elliptic equations in bounded quasiconvex Lipschitz domains. Our result is new even for Laplace operator in convex domains.
We study the interior nodal sets, $Z_λ$ of Steklov eigenfunctions in an $n$-dimensional relatively compact manifolds $M$ with boundary and show that one has the lower bounds $|Z_λ|\ge cλ^{\frac{2-n}2}$ for the size of its $(n-1)$-dimensional Hausdorff measure. The proof is based on a Dong-type identity and estimates for the gradient of Steklov eigenfunctions, similar to those in \cite{SZ1} and \cite{SZ2}, respectively.
Let $Ω$ be a bounded annular $C^{1,1}$ domain in $\mathbb{R}^2$ which is left invariant under the action of the dihedral group $D_n$ of isometries of $\mathbb{R}^2$ .We show that the nodal line of a second Dirichlet eigenfunction must intersect the boundary of $Ω$, under suitable conditions on $\frac{\partial}{\partial θ}$ .
Nodal Deficiency of Neumann Eigenfunctions on a Symmetric Dumbbell Domain
研究对称哑铃域上Neumann特征函数的节点缺陷，揭示其随颈部宽度变化的行为，并提供判断零节点缺陷特征函数的准则。
We study the size of nodal sets of Laplacian eigenfunctions on compact Riemannian manifolds without boundary and recover the currently optimal lower bound by comparing the heat flow of the eigenfunction with that of an artifically constructed diffusion process. The same method should apply to a number of other questions; for example, we prove a sharp result saying that a nodal domain cannot be entirely contained in a small neighbourhood of a 'reasonably flat' surface. We expect the arising concepts to have more connections to classical theory and pose some conjectures in that direction.
We show that Steklov eigenfunctions in a bounded Lipschitz domain have wavelength dense nodal sets near the boundary, in contrast to what can happen deep inside the domain. As a converse, in a two-dimensional Lipschitz domain $Ω$, we prove that any nodal domain of a Steklov eigenfunction contains a half-ball centered at $\partialΩ$ of radius $c_Ω/λ$.
We investigate the geometric properties of Steklov eigenfunctions in smooth manifolds. We derive the refined doubling estimates and Bernstein's inequalities. For the real analytic manifolds, we are able to obtain the sharp upper bound for the measure of interior nodal sets.
We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in $\mathbb{R}^n$ - the eigenfunctions of the Dirichlet-to-Neumann map. Under the assumption that the domain $Ω$ is $C^2$, we prove a doubling property for the eigenfunction $u$. We estimate the Hausdorff $\mathcal H^{n-2}$-measure of the nodal set of $u|_{\partial Ω}$ in terms of the eigenvalue $λ$ as $λ$ grows to infinity. In case that the domain $Ω$ is analytic, we prove a polynomial bound O($λ^6$). Our arguments, which make heavy use of Almgren's frequency functions, are built on the previous wo…
Graph structure of the nodal set and bounds on the number of critical points of eigenfunctions on Riemannian manifolds
本文将紧致二维Riemannian流形上Laplacian特征函数的零点集识别为嵌入度量图，并利用初等图论工具，依据特征值指标$k$与曲面Euler-Poincaré示性数估计了特征函数零点集的临界点数量及消失阶之和。
In this paper we consider the problem of prescribing the nodal set of low-energy eigenfunctions of the Laplacian. Our main result is that, given any separating closed hypersurface Σin a compact n-manifold M, there is a Riemannian metric on M such that the nodal set of its first nontrivial eigenfunction is Σ. We present a number of variations on this result, which enable us to show, in particular, that the first nontrivial eigenfunction can have as many non-degenerate critical points as one wishes.
We construct a multiply connected domain in $\mathbb{R}^2$ for which the second eigenfunction of the Laplacian with Robin boundary conditions has an interior nodal line. In the process, we adapt a bound of Donnelly-Fefferman type to obtain a uniform estimate on the size of the nodal sets of a sequence of solutions to a certain class of elliptic equations in the interior of a sequence of domains, which does not depend directly on any boundary behaviour. This also gives a new proof of the nodal line property of the example in the Dirichlet case.
The nodal set of the Laplacian eigenfunction has co-dimension one and has finite hypersurface measure on a compact Riemannian manifold. In this paper, we investigate the distribution of the nodal sets of eigenfunctions, when the metric on the manifold is analytic. We prove that if the eigenfunctions are equidistributed at a small scale, then the weak limits of the hypersurface volume form of their nodal sets are comparable to the volume form on the manifold. In particular, on the negatively curved manifolds with analytic metric and on the tori, we show that in any eigenbasis, there is a full d…
In this note, we discuss a question posed by T. Hoffmann-Ostenhof concerning the parity of the number of nodal domains for a non-constant eigenfunction of the Laplacian on flat tori. We present two results. We first show that on the torus $(\mathbb{R}/2π\mathbb{Z})^{2}$, a non-constant eigenfunction has an even number of nodal domains. We then consider the torus $(\mathbb{R}/2π\mathbb{Z})\times(\mathbb{R}/2ρπ\mathbb{Z})\,$, with $ρ=\frac{1}{\sqrt{3}}\,$, and construct on it an eigenfunction with three nodal domains.
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。
Nodal sets of eigenfunctions of sub-Laplacians
本文首次研究紧流形上次拉普拉斯算子特征函数的节点集，揭示其各向异性行为，并给出反例说明其豪斯多夫测度不能被 Yau 猜想的 $\sqrtλ$ 所界定。
We investigate the interior nodal sets $\mathcal{N}_λ$ of Steklov eigenfunctions on connected and compact surfaces with boundary. The optimal vanishing order of Steklov eigenfunctions is shown be $Cλ$. The singular sets $\mathcal{S}_λ$ are finite points on the nodal sets. We are able to prove that the Hausdorff measure $H^0(\mathcal{S}_λ)\leq Cλ^2$. Furthermore, we obtain an upper bound for the measure of interior nodal sets $H^1(\mathcal{N}_λ)\leq Cλ^{\frac{3}{2}}$. Here those positive constants $C$ depend only on the surfaces.
This article concerns the number of nodal domains of eigenfunctions of the Laplacian on special Riemannian $3$-manifolds, namely nontrivial principal $S^1$ bundles $P \to X$ over Riemann surfaces equipped with certain $S^1$ invariant metrics, the Kaluza-Klein metrics. We prove for generic Kaluza-Klein metrics that any Laplacian eigenfunction has exactly two nodal domains unless it is invariant under the $S^1$ action. We also construct an explicit orthonormal eigenbasis on the flat $3$-torus $\mathbb{T}^3$ for which every non-constant eigenfunction belonging to the basis has two nodal domains.
We prove sharp upper and lower bounds for the nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces with boundary. The argument involves frequency function methods for harmonic functions in the interior of the surface as well as the construction of exponentially accurate approximations for the Steklov eigenfunctions near the boundary.
We study the nodal set of Laplace eigenfunctions on the flat $2d$ torus $\mathbb{T}^2$. We prove an asymptotic law for the nodal length of such eigenfunctions, under some growth assumptions on their Fourier coefficients. Moreover, we show that their nodal set is asymptotically equidistributed on $\mathbb{T}^2$. The proofs are based on Bourgain's de-randomisation technique and the main new ingredient, which might be of independent interest, is the integrability of arbitrarily large powers of the doubling index of Laplace eigenfunctions on $\mathbb{T}^2$, based on the work of Nazarov \cite{N93,N…
Nodal solutions of a parametric (p_1,p_2)-Laplacian system, with Neumann boundary conditions, are obtained by chiefly constructing appropriate sub-super-solution pairs.
We study the stability properties of nodal sets of Laplace eigenfunctions on compact manifolds under specific small perturbations. We prove that nodal sets are fairly stable if said perturbations are relatively small, more formally, supported at a sub-wavelength scale. We do not need any assumption on the topology of the nodal sets. As an indirect application, we are able to show that a certain "Payne property" concerning the second nodal line remains stable under controlled perturbations of the domain.
We give uniform upper and lower bounds for the L^2 norm of the restriction of eigenfunctions of the Laplacian on the three-dimensional standard flat torus to surfaces with non-vanishing curvature. We also present several related results concerning the nodal sets of eigenfunctions.
Hessian estimates for Dirichlet and Neumann eigenfunctions of Laplacian
本文借助 Riemannian 流形上的随机分析，建立带边界紧流形中 Laplacian 的 Dirichlet 特征函数 Hessian 上下界及尖锐增长估计，并推广至 Neumann 情形。
We prove a result, announced by F. Nazarov, L. Polterovich and M. Sodin that exhibits a relation between the average local growth of a Laplace eigenfunction on a closed surface and the global size of its nodal set. More precisely, we provide a lower and an upper bound to the Hausdorff measure of the nodal set in terms of the expected value of the growth exponent of an eigenfunction on disks of wavelength like radius. Combined with Yau's conjecture, the result implies that the average local growth of an eigenfunction on such disks is bounded by constants in the semi-classical limit. We also obt…
In this paper we are interested in the following fourth order eigenvalue problem coming from the buckling of thin films on liquid substrates: \begin{equation*} \begin{cases} Δ^2 u+ κ^2 u=-λΔu &\text{in } B_1,\newline u=\partial_r u= 0 &\text{on } \partial B_1, \end{cases} \end{equation*} where $B_1$ is the unit ball in $\mathbb{R}^N$. When $κ> 0$ is small, we show that the first eigenvalue is simple and the first eigenfunction, which gives the shape of the film for small displacements, is positive. However, when $κ$ increases, we establish that the first eigenvalue is not always simple and the…
We consider a nonlinear Neumann problem driven by a $p$-Laplacian-type, nonhomogeneous elliptic differential operator and a Carathéodory reaction term. In this paper we prove the existence of two extremal constant sign smooth solutions and a nontrivial nodal smooth solution. In the proof we use variational methods with truncation techniques, critical point theory and Morse theory (critical groups).
This paper is a continuation and an extension of our recent work [3] on the geometric structures of Laplacian eigenfunctions and their applications to inverse scattering problems. In [3], the analytic behaviour of the Laplacian eigenfunctions is investigated at a point where two nodal or generalised singular lines intersect. It reveals a certain intriguing property that the vanishing order of the eigenfunction at the intersecting point is related to the rationality of the intersecting angle. In this paper, we consider the 3D counterpart of such a study by studying the analytic behaviours of th…
This note concerns the nodal sets of eigenfunctions of semiclassical Schrödinger operators acting on compact, smooth, Riemannian manifolds, with no boundary. We prove that if H is a separating hypersurface that lies inside the classically forbidden region, then H cannot persist as a component of the zero set of infinitely many eigenfunctions. In addition, on real analytic surfaces, we obtain sharp upper bounds for the number of intersections of the zero sets of the Schrödinger eigenfunctions with a fixed curve that lies inside the classically forbidden region.
The sharp estimate of nodal sets for Dirichlet Laplace eigenfunctions in polytopes
本文研究有界 Lipschitz 多面体中 Dirichlet Laplace 特征函数的节点集测度，证明其上界为 $C(P)\sqrtλ$，并将已有拟凸区域结果推广至一般多面体。
We study the nodal curves of low energy Dirichlet eigenfunctions in generalized curvilinear quadrilaterals. The techniques can be seen as a generalization of the tools developed by Grieser-Jerison in a series of works on convex planar domains and rectangles with one curved edge and a large aspect ratio. Here, we study the structure of the nodal curve in greater detail, in that we find precise bounds on its curvature, with uniform estimates up to the two points where it meets the domain at right angles, and show that many of our results hold for relatively small aspect ratios of the side length…
In this article, we consider eigenfunctions $u$ of the bi-harmonic operator, i.e., $\triangle^2u=λ^2u$ on $Ω$ with some homogeneous linear boundary conditions. We assume that $Ω\subseteq\mathbb{R}^n$ ($n\geq2$) is a $C^{\infty}$ bounded domain, $\partialΩ$ is piecewise analytic and $\partialΩ$ is analytic except a set $Γ\subseteq\partialΩ$ which is a finite union of some compact $(n-2)$ dimensional submanifolds of $\partialΩ$. The main result of this paper is that the measure upper bounds of the nodal sets of the eigenfunctions is controlled by $\sqrtλ$. We first define a frequency function an…
Eigenfunction asymptotics and nodal domain estimates for dumbbell domains
研究哑铃区域Neumann特征函数节点结构，证明颈部变薄时节点域数量增加，第三特征函数达到Courant上界，并利用匹配渐近分析克服奇异极限困难。
We consider the problem of prescribing the nodal set of the first nontrivial eigenfunction of the Laplacian in a conformal class. Our main result is that, given a separating closed hypersurface $Σ$ in a compact Riemannian manifold $(M,g_0)$ of dimension $d \geq 3$, there is a metric $g$ on $M$ conformally equivalent to $g_0$ and with the same volume such that the nodal set of its first nontrivial eigenfunction is a $C^0$-small deformation of $Σ$ (i.e., $Φ(Σ)$ with $Φ: M \to M$ a diffeomorphism arbitrarily close to the identity in the $C^0$ norm).
In a recent paper, we exhibit a link between the average local growth of Laplace eigenfunctions on surfaces and the size of their nodal set. In that paper, the average local growth is computed using the uniform - or $L^\infty$ - growth exponents on disks of wavelength radius. The purpose of this note is to prove similar results for a broader class of $L^q$ growth exponents with $q \in (1, \infty)$. More precisely, we show that the size of the nodal set is bounded above and below by the product of the average local $L^q$ growth with the frequency. We briefly discuss the relation between this ne…
The Sharp Measure Upper Bound of the Nodal Sets of Neumann Laplace Eigenfunctions on C1,1 Domains
本文研究了 C^{1,1} 区域上 Neumann Laplace 特征函数的结集测度问题，证明了其零集的 (n-1) 维 Hausdorff 测度不超过 C√λ 的精确上界。
Existence and location of nodal solutions for quasilinear convection-absorption Neumann problems
本文研究了拟线性对流-吸收Neumann问题，利用子-超解技术与单调算子理论证明了结点解与常号解的存在性，并通过常号解确立了结点解的位置原理。
In this paper, we obtain the upper bounds for the Hausdorff measures of nodal sets of eigenfunctions with the Robin boundary conditions, i.e., \begin{equation*} {\left\{\begin{array}{l} \triangle u+λu=0,\quad in\quad Ω,\\ u_ν+μu=0,\quad on\quad\partialΩ, \end{array} \right.} \end{equation*} where the domain $Ω\subseteq\mathbb{R}^n$, $u_ν$ means the derivative of $u$ along the outer normal direction of $\partialΩ$. We show that, if $Ω$ is bounded and analytic, and the corresponding eigenvalue $λ$ is large enough,then the measure upper bounds for the nodal sets of eigenfunctions are $C\sqrtλ$, w…
In this short note we show that the lower bounds of Mangoubi on the inner radius of nodal domains can be improved for quantum ergodic sequences of eigenfunctions, according to a certain power of the radius of shrinking balls on which the eigenfunctions equidistribute. We prove such improvements using a quick application of our recent results (arXiv:1606.02057), which give modified growth estimates for eigenfunctions that equidistribute on small balls. Since by the results of Han and Hezari-Rivière small scale QE holds for negatively curved manifolds on logarithmically shrinking balls, we get l…
Let $Ω$ be a bounded domain in $\mathbb{R}^n$ with $C^{1}$ boundary and let $u_λ$ be a Dirichlet Laplace eigenfunction in $Ω$ with eigenvalue $λ$. We show that the $(n-1)$-dimensional Hausdorff measure of the zero set of $u_λ$ does not exceed $C(Ω)\sqrtλ$. This result is new even for the case of domains with $C^\infty$-smooth boundary.
Nodal sets and continuity of eigenfunctions of Kre\uı-Feller operators
该研究针对$\mathbb{R}^{d}$上紧支集正有限Borel测度下的Kre\u012d-Feller算子，证明其连续特征函数的节点集分域数量范围，并推广经典拉普拉斯算子节点集定理，同时证得存在经典格林函数的有界域上特征函数连续。
Hausdorff measure bounds for nodal sets of Steklov eigenfunctions
本文研究有界$\mathcal{C}^2$区域内Steklov特征函数的节点集，建立了与特征值无关的Hausdorff测度下界，并给出含$\lambda\log(\lambda+e)$因子的近乎精确上界。
We build a one-parameter family of S^{1}-invariant metrics on the unit disc with fixed total area for which the second eigenvalue of the Laplace operator in the case of both Neumann and Dirichlet boundary conditions is simple and has an eigenfunction with a closed nodal line. In the case of Neumann boundary conditions, we also prove that this eigenfunction attains its maximum at an interior point, and thus provide a counterexample to the hot spots conjecture on a simply connected surface. This is a consequence of the stronger result that within this family of metrics any given (finite) number …
Neumann's nodal line may be closed on doubly-connected planar domains
研究证明在具有一个孔的平面域中，Neumann特征函数的节点线可完全闭合于域内，改进了关于特征函数收敛性的已知结果。
We consider two-dimensional Schrödinger operators in bounded domains. We analyze relations between nodal domains of eigenfunctions, spectral minimal partitions and spectral properties of the corresponding operator. The main results concern the existence and regularity of the minimal partitions associated with the nodal sets as the nodal domains of Courant-sharp eigenfunctions.
We show that, in round spheres of dimension $n\geq3$, for any given collection of codimension 2 smooth submanifolds $\mathfrak{S}:=\{Σ_1,...,Σ_N\}$ of arbitrarily complicated topology ($N$ being the complex dimension of the spinor bundle), there is always an eigenfunction $ψ=(ψ_1,...,ψ_N)$ of the Dirac operator such that each submanifold $Σ_a$, modulo ambient diffeomorphism, is a structurally stable nodal set of the spinor component $ψ_a$. The result holds for any choice of trivialization of the spinor bundle. The emergence of these structures takes place at small scales and sufficiently high …
A Counterexample to the Closed Nodal-Line Conjecture for the Third Eigenfunction of the Planar Dirichlet Laplacian
针对平面Dirichlet Laplacian第三本征函数的闭节线猜想，构造出一个第三Dirichlet特征值为单重、节集含两条不相交闭曲线的反例，完成猜想的否定。
It is an open problem in general to prove that there exists a sequence of $Δ_g$-eigenfunctions $ϕ_{j_k}$ on a Riemannian manifold $(M, g)$ for which the number $N(ϕ_{j_k}) $ of nodal domains tends to infinity with the eigenvalue. Our main result is that $N(ϕ_{j_k}) \to \infty$ along a subsequence of eigenvalues of density $1$ if the $(M, g)$ is a non-positively curved surface with concave boundary, i.e. a generalized Sinai or Lorentz billiard. Unlike the recent closely related work of Ghosh-Reznikov-Sarnak and of the authors on the nodal domain counting problem, the surfaces need not have any …
The width of a convex curve in the plane is the minimal distance between a pair of parallel supporting lines of the curve. In this paper we study the width of nodal lines of eigenfunctions of the Laplacian on the standard flat torus. We prove a variety of results on the width, some having stronger versions assuming a conjecture of Cilleruelo and Granville asserting a uniform bound for the number of lattice points on the circle lying in short arcs.
Hausdorff measure bound for the nodal sets of Neumann Laplace eigenfunctions
该研究针对带$\mathcal{C}^{1,1}$边界的有界域内的Neumann拉普拉斯特征函数，推导得出其节点集的$n-1$维Hausdorff测度满足$\mathcal{H}^{n-1}(\{u_\lambda=0\})\leq C\sqrt{\lambda}$的上界估计。
In this paper, we show that equality in Courant's nodal domain theorem can only be reached for a finite number of eigenvalues of the Neumann Laplacian, in the case of an open, bounded and connected set in R n with a C 1,1 boundary. This result is analogous to Pleijel's nodal domain theorem for the Dirichlet Laplacian (1956). It confirms, in all dimensions, a conjecture formulated by Pleijel, which had already been solved by I. Polterovich for a two-dimensional domain with a piecewise-analytic boundary (2009). We also show that the argument and the result extend to a class of Robin boundary con…
The paper deals with asymptotic nodal geometry for the Laplace-Beltrami operator on closed surfaces. Given an eigenfunction f corresponding to a large eigenvalue, we study local asymmetry of the distribution of sign(f) with respect to the surface area. It is measured as follows: take any disc centered at the nodal line {f=0}, and pick at random a point in this disc. What is the probability that the function assumes a positive value at the chosen point? We show that this quantity may decay logarithmically as the eigenvalue goes to infinity, but never faster than that. In other words, only a mil…
We study the nodal sets of eigenfunctions of the Laplacian on the standard d-dimensional flat torus. The question we address is: Can a fixed hypersurface lie on the nodal sets of eigenfunctions with arbitrarily large eigenvalue? In dimension two, we show that this happens only for segments of closed geodesics. In higher dimensions, certain cylindrical sets do lie on nodal sets corresponding to arbitrarily large eigenvalues. Our main result is that this cannot happen for hypersurfaces with nonzero Gauss-Kronecker curvature. In dimension two, the result follows from a uniform lower bound for the…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。