Mourad Choulli, Laurent Kayser
We adapt in the present note the perturbation method introduced in [3] to get a Gaussian lower bound for the Neumann heat kernel of the Laplace-Beltrami operator on an open subset of a compact Riemannian manifold.
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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 $\mathbb{S}^N$.
Refinements to the Kuznetsov Formula for Laplace-Beltrami Operators
研究拉普拉斯 - 贝尔特拉米算子的库兹涅佐夫公式，通过改进怀曼和奚的结果，在余维数为一的情况去掉常数，在环回时间集测度为零的情况证明新渐近性。
We present a boundary integral method, and an accompanying boundary element discretization, for solving boundary-value problems for the Laplace-Beltrami operator on the surface of the unit sphere $§$ in $\mathbb{R}^3$. We consider a closed curve ${\cal C}$ on ${\cal S}$ which divides ${\cal S}$ into two parts ${\cal S}_1$ and ${\cal S}_2$. In particular, ${\cal C} = \partial {\cal S}_1$ is the boundary curve of ${\cal S}_1$. We are interested in solving a boundary value problem for the Laplace-Beltrami operator in $§_2$, with boundary data prescribed on $\C$.
Anisotropic Calderón problem of a nearly Laplace-Beltrami operator of order $2+$
本文研究闭黎曼流形上对数拉普拉斯算子的各向异性 Calderón 问题，证明柯西数据集可在标准规范等价意义下唯一恢复流形几何。
In this paper we prove a concentration result and an error estimate for a model of heat conduction in composite materials having a microscopic structure arranged in a perodic array and thermally active membranes separating the heat conductive phases.
We study a concentration and homogenization problem modelling electrical conduction in a composite material. The novelty of the problem is due to the specific scaling of the physical quantities characterizing the dielectric component of the composite. This leads to the appearance of a peculiar displacement current governed by a Laplace-Beltrami pseudo-parabolic equation. This pseudo-parabolic character is present also in the homogenized equation, which is obtained by the unfolding technique.
We investigate the mixed Dirichlet-Neumann boundary value problems for the Laplace-Beltrami equation on a smooth hypersurface $\mathcal{C}$ with the smooth boundary in non-classical setting in the Bessel potential spaces $\mathbb{H}^1_p(\mathcal{C})$ for $1<p<\infty$. To the initial BVP we apply quasilocalization and obtain model BVPs for the Laplacian. The model mixed BVP on the half plane is investigated by potential method and is reduced to an equivalent system of Mellin convolution equations in Bessel potential and Besov spaces. The symbol of the obtained system is written explicitly, whic…
In this article we prove $L^p$ estimates for resolvents of Laplace-Beltrami operators on compact Riemannian manifolds, generalizing results of Kenig, Ruiz and Sogge in the Euclidean case and Shen for the torus. We follow Sogge and construct Hadamard's parametrix, then use classical boundedness results on integral operators with oscillatory kernels related to the Carleson and Sjölin condition. Our initial motivation was to obtain $L^p$ Carleman estimates with limiting Carleman weights generalizing those of Jerison and Kenig; we illustrate the pertinence of $L^p$ resolvent estimates by showing t…
Resolvent estimates in Schatten spaces for Laplace-Beltrami operators on compact manifolds
本文研究了紧流形上拉普拉斯-贝尔特里算子在临界指数处的施图森空间中的解析估计，通过已知的哈达玛参数化估计界进行证明。
The paper is pertaining to the spectral theory of operators and boundary value problems for differential equations on manifolds. Eigenvalues of such problems are studied as functionals on the space of domains. Resolvent continuity of the corresponding operators is established under domain deformation and estimates of continuity moduli of their eigenvalues $\slash$ eigenfunctions are obtained provided the boundary of nonperturbed domain is locally represented as a graph of some continuous function and domain deformation is measured with respect to the Hausdorff--Pompeiu metric.
Stability of determining a Dirichlet-Laplace-Beltrami operator from its boundary spectral data
本文针对由边界谱数据确定Dirichlet-Laplace-Beltrami算子的反问题，在完整与部分谱数据两种情形下建立了稳定性不等式。
Plane-wave representation for the Laplace--Beltrami equation on a sphere. Application to the Green's function
本文提出了一种扩展的平面波表示法，用于球面上的波场，通过研究复化球面的几何和拉普拉斯-贝尔特拉米算子的性质，扩展了平面波表示法的适用范围，并通过研究实球面上的格林函数来展示方法。
Given a compact Riemannian manifold $(M, g)$ without boundary, we estimate the Lebesgue norm of Laplace-Beltrami eigenfunctions when restricted to a wide variety of subsets $Γ$ of $M$. The sets $Γ$ that we consider are Borel measurable, Lebesgue-null but otherwise arbitrary with positive Hausdorff dimension. Our estimates are based on Frostman-type ball growth conditions for measures supported on $Γ$. For large Lebesgue exponents $p$, these estimates provide a natural generalization of $L^p$ bounds for eigenfunctions restricted to submanifolds, previously obtained in \cite{Ho68, Ho71, Sog88, B…
We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.
We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifold with boundary. These domains are close to half balls of small radius centered at a nondegenerate critical point of the mean curvature function of the boundary of the manifold, and their boundary intersects the boundary of the manifold orthogonally.
For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt-Caffarelli-Friedman and Caffarelli-Jerison-Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the Laplace-Beltrami operator on Riemannian manifolds. As an application we show that our monotonicity theorems can be employed to prove the Lipschitz continuity for the solutions of a general class of two-phase free boundary problems on Riemannian manifolds.
Let $\De u+\la u=\De v+\la v=0$, where $\De$ is the Laplace--Beltrami operator on a compact connected smooth manifold $M$ and $\la>0$. If $H^1(M)=0$ then there exists $p\in M$ such that $u(p)=v(p)=0$. For homogeneous $M$, $H^1(M)\neq0$ implies the existence of a pair $u,v$ as above that has no common zero.
We give estimates for the $L^p$ norm ($2\leq p \leq +\infty$) of the restriction to a curve of the eigenfunctions of the Laplace Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere these estimates are sharp. If the curve has non vanishing geodesic curvature, we can improve our results. We also show how our approach apply to higher dimensional manifolds.
The problem of determining the domain of the closure of the Laplace-Beltrami operator on a 2D almost-Riemannian manifold is considered. Using tools from theory of Lie groupoids natural domains of perturbations of the Laplace-Beltrami operator are found. The main novelty is that the presented method allows us to treat geometries with tangency points. This kind of singularity is difficult to treat since those points do not have a tubular neighbourhood compatible with the almost-Riemannian metric.
In this paper we provide an integral representation of the fractional Laplace-Beltrami operator for general riemannian manifolds which has several interesting applications. We give two different proofs, in two different scenarios, of essentially the same result. One of them deals with compact manifolds with or without boundary, while the other approach treats the case of riemannian manifolds without boundary whose Ricci curvature is uniformly bounded below.
Eigenfunction equivalence for the fractional Laplace-Beltrami operator and the classical Helmholtz equation
本文建立了分数次Laplace-Beltrami算子与经典Helmholtz方程的本征函数等价性，基于Seeley的椭圆算子复幂构造与伪微分符号演算完成证明，并将其应用于固定频率逆散射问题。
In this paper we deal with the problems of the weak localization of the eigenfunction expansions related to Laplace-Beltrami operator on unit sphere. The conditions for weak localization of Fourier-Laplace series are investigated by comparing the Riesz and Cesaro methods of summation for eigenfunction expansions of the Laplace-Beltrami operator. It is shown that the weak localization principle for the integrable functions f(x) at the point x depends not only on behavior of the function around x but on the behavior of the function around diametrically opposite point \overline{x}.
An interface formulation of the Laplace-Beltrami problem on piecewise smooth surfaces
该研究建立分片光滑曲面上Laplace-Beltrami算子的$L^2$可逆性理论，将问题弱形式重构为带棱边连续性条件的界面问题，并用旋转曲面高阶数值算例验证，为相关物理问题求解提供理论与数值基础。
We present a simple, short and elementary proof that if $v$ is a Beltrami flow with a finite energy in $\mathbb R^3$ then $v=0$. In the case of the Beltrami flows satisfying $v\in L^\infty _{loc} (\Bbb R^3) \cap L^q(\Bbb R^3)$ with $q\in [2, 3)$, or $|v(x)|=O(1/|x|^{1+\varepsilon})$ for some $\varepsilon >0$, we provide a different, simple proof that $v=0$.
We introduce the notion of a family of convolution operators associated with a given elliptic partial differential operator. Such a convolution structure is shown to exist for a general class of Laplace-Beltrami operators on two-dimensional manifolds endowed with cone-like metrics. This structure gives rise to a convolution semigroup representation for the Markovian semigroup generated by the Laplace-Beltrami operator. In the particular case of the operator $\mathcal{L} = \partial_x^2 + {1 \over 2x} \partial_x + {1 \over x} \partial_θ^2$ on $\mathbb{R}^+ \times \mathbb{T}$, we deduce the exist…
We study the spectrum of the Laplace-Beltrami operator on ellipsoids. For ellipsoids that are close to the sphere, we use analytic perturbation theory to estimate the eigenvalues up to two orders. We show that for biaxial ellipsoids sufficiently close to the sphere, the first $N$ eigenvalues have multiplicity at most two, and characterize those that are simple. For the triaxial ellipsoids sufficiently close to the sphere that are not biaxial, we show that at least the first sixteen eigenvalues are all simple. We also give the results of various numerical experiments, including comparisons to o…
We study a discrete model of the Laplacian in $\mathbb{R}^2$ that preserves the geometric structure of the original continual object. This means that, speaking of a discrete model, we do not mean just the direct replacement of differential operators by difference ones but also a discrete analog of the Riemannian structure. We consider this structure on the appropriate combinatorial analog of differential forms. Self-adjointness and boundness for a discrete Laplacian are proved. We define the Green function for this operator and also derive an explicit formula of the one.
We study the evolution of the heat and of a free quantum particle (described by the Schrödinger equation) on two-dimensional manifolds endowed with the degenerate Riemannian metric $ds^2=dx^2+|x|^{-2α}dθ^2$, where $x\in \mathbb R$, $θ\in\mathbb T$ and the parameter $α\in\mathbb R$. For $α\le-1$ this metric describes cone-like manifolds (for $α=-1$ it is a flat cone). For $α=0$ it is a cylinder. For $α\ge 1$ it is a Grushin-like metric. We show that the Laplace-Beltrami operator $Δ$ is essentially self-adjoint if and only if $α\notin(-3,1)$. In this case the only self-adjoint extension is the F…
We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a $m$-dimensional submanifold $M$ in $R^d$ as the sample size $n$ increases and the neighborhood size $h$ tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of $O\Big(\big(\frac{\log n}{n}\big)^\frac{1}{2m}\Big)$ to the eigenvalues and eigenfunctions of the weighted Laplace-Beltrami operator of $M$. No information on the submanifold $M$ is needed in the construction of the graph or the "out-of-sample extension" of the eigenvectors. Of …
Spectral estimate for the Laplace-Beltrami operator on the hyperbolic half-plane
本文在双曲半平面上证明了谱投影器的自然最优不确定性原理，为非欧几里得几何环境下的高频分析提供了新贡献。
On the concentration of the Fourier coefficients for products of Laplace-Beltrami eigenfunctions on real-analytic manifolds
本文研究了闭解析流形上 Laplace-Beltrami 特征函数乘积的 Fourier 系数，证明了在特征值满足特定条件时系数呈指数衰减，并指出仅需有限个系数即可恢复 99% 的 $L^2$-质量。
The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on a flat disc than on any other surface of revoltuion immersed in Euclidean space with the same boundary.
We investigate the regularity of semi-stable, radially symmetric, and decreasing solutions for a class of quasilinear reaction-diffusion equations in the inhomogeneous context of Riemannian manifolds. We prove uniform boundedness, Lebesgue and Sobolev estimates for this class of solutions for equations involving the p-Laplace Beltrami operator and locally Lipschitz non-linearity. We emphasize that our results do not depend on the boundary conditions and the specific form of the non-linearities and metric. Moreover, as an application, we establish regularity of the extremal solutions for equati…
Let $(M,g)$ be a connected, closed, orientable Riemannian surface and denote by $λ_k(M,g)$ the $k$-th eigenvalue of the Laplace-Beltrami operator on $(M,g)$. In this paper, we consider the mapping $(M, g)\mapsto λ_k(M,g)$. We propose a computational method for finding the conformal spectrum $Λ^c_k(M,[g_0])$, which is defined by the eigenvalue optimization problem of maximizing $λ_k(M,g)$ for $k$ fixed as $g$ varies within a conformal class $[g_0]$ of fixed volume $textrm{vol}(M,g) = 1$. We also propose a computational method for the problem where $M$ is additionally allowed to vary over surfac…
It is known (E.L. Green (1997), O. Post (2003)) that for an arbitrary $m\in\mathbb{N}$ one can construct a periodic non-compact Riemannian manifold $M$ with at least $m$ gaps in the spectrum of the corresponding Laplace-Beltrami operator $-Δ_M$. In this work we want not only to produce a new type of periodic manifolds with spectral gaps but also to control the edges of these gaps. The main result of the paper is as follows: for arbitrary pairwise disjoint intervals $(\a_j,\b_j)\subset[0,\infty)$, $j=1,...,m$ ($m\in\mathbb{N}$), for an arbitrarily small $δ>0$ and for an arbitrarily large $L>0$ …
The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on an annulus than on any other surface of revolution in $\mathbb{R}^3$ with the same boundary. This is established by defining a sequence of shrinking cylinders about the axis of symmetry and proving that flattening a surface outside of each cylinder successively increases the eigenvalues. A similar argument shows that the Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on a half-helicoid than on any other screw surface in $\mathbb{R}^2 \times \mathbb{S}^1$ with the same boundary.
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 a compact Kähler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eigenspaces of the Laplace-Beltrami operator. Because of the high degree of symmetry the Laplace-Beltrami operator on forms can not be quantum ergodic. We show that after taking these symmetries into account quantum ergodicity holds for the Laplace-Beltrami operator and for the Spin…
In this paper we continue the study on the resolvent estimates of the Laplace-Beltrami operator $Δ_g$ on a compact manifolds $M$ with dimension $n\geq3$. On the Sobolev line $1/p-1/q=2/n$ we can prove that the resolvent $(Δ_g+ζ)^{-1}$ is uniformly bounded from $L^p$ to $L^q$ when $(p,q)$ are within the admissible range $p\leq2(n+1)/(n+3)$ and $q\geq2(n+1)/(n-1)$ and $ζ$ is outside a parabola opening to the right and a small disk centered at the origin. This naturally generalizes the previous results in \cite{Kenig} and \cite{bssy} which addressed only the special case when $p=2n/(n+2), q=2n/(n…
Let $e(x,y,ł)$ be the spectral function and $χ_ł$ the unit band spectral projection operator, with respect to the Laplace-Beltrami operator $\D_M$ on a closed Riemannian manifold $M$. We firstly review the one-term asymptotic formula of $e(x,x,ł)$ as $ł\to\infty$ by H{\" o}rmander (1968) and the one of $\p^\al_x\p^\bt_y e(x,y,ł)|_{x=y}$ as $ł\to\infty$ in a geodesic normal coordinate chart by the author (2004) and the sharp asymptotic estimates from above of the mapping norm $\|χ_ł\|_{L_2\to L_p}$ ($2\leq p\leq\infty$) by Sogge (1988 $&$ 1989) and of the mapping norm $\|χ_ł\|_{L_2\to {\rm Sobo…
We consider the inverse boundary value problem for the first order perturbation of the polyharmonic operator $\mathcal L_{g,X,q}$, with $X$ being a $W^{1,\infty}$ vector field and $q$ being an $L^\infty$ function on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that the knowledge of the Dirichlet-to-Neumann determines $X$ and $q$ uniquely. The method is based on the construction of complex geometrical optics solutions using the Carleman estimate for the Laplace-Beltrami operator due to Dos Santos Ferr…
We give necessary and sufficient conditions for the solvability of some semilinear elliptic boundary value problems involving the Laplace operator with linear and nonlinear highest order boundary conditions involving the Laplace-Beltrami operator.
Random acoustic boundary conditions and Weyl's law for Laplace-Beltrami operators on non-smooth boundaries
本文研究了非光滑边界上Laplace-Beltrami算子的随机边界条件，通过特征函数构造随机化方法，揭示残差紧致性与Weyl定律的关系。
Self-adjointness criteria and self-adjoint extensions of the Laplace-Beltrami operator on $α$-Grushin manifolds
本文针对α-Grushin流形上Laplace-Beltrami算子的自伴性问题，引入适应奇异几何的伪微分算子演算，给出曲率Laplacian本质自伴性判据，并在非本质自伴情形下构造自然自伴延拓，结果推广至该演算下椭圆的一大类微分算子。
We obtain a sharp lower estimate on eigenvalues of Laplace--Beltrami operator on a hyperbolic surface with injectivity radius bounded from the below.
Consider an $L^2$-normalized Laplace-Beltrami eigenfunction $e_λ$ on a compact, boundary-less Riemannian manifold with $Δe_λ= -λ^2 e_λ$. We study eigenfunction triple products \[ \langle e_λe_μ, e_ν\rangle = \int e_λe_μ\overline{e_ν} \, dV. \] We show the overall $\ell^2$-concentration of these triple products is determined by the measure of some set of configurations of triangles with side lengths equal to the frequencies $λ,μ,$ and $ν$. A rapidly vanishing proportion of this mass lies in the `classically forbidden' regime where $λ, μ,$ and $ν$ fail to satisfy the triangle inequality. As a co…
The critical case for the concentration of eigenfunctions on singular Riemannian manifolds
本文研究带边界的奇异黎曼流形上Laplace-Beltrami算子特征函数的集中现象，在临界情形下证明其平均密度分布于所有长度尺度，并给出精确描述。
Minimax Rates for the Estimation of Eigenpairs of Weighted Laplace-Beltrami Operators on Manifolds
本文研究了流形上椭圆算子特征对估计的极小化极大速率，证明在 $H^1(M)$ 意义下逼近特征对的速率为 $n^{-2/(d+4)}$，并指出图拉普拉斯算子特征对在稍强正则性下可达到匹配该下界的逼近误差，从而在更强范数和一致收敛性上扩展现有图学习理论。
We consider the conjecture proposed in Matsumoto, Zhang and Schiebinger (2022) suggesting that optimal transport with quadratic regularisation can be used to construct a graph whose discrete Laplace operator converges to the Laplace--Beltrami operator. We derive first order optimal potentials for the problem under consideration and find that the resulting solutions exhibit a surprising resemblance to the well-known Barenblatt--Prattle solution of the porous medium equation. Then, relying on these first order optimal potentials, we derive the pointwise $L^2$-limit of such discrete operators bui…
On the intertwining map between Coulomb and hyperbolic scattering
本文提出一个将库仑散射与双曲散射间本征函数空间联系起来的单位元算子，推广了福克变换到连续谱，并解释了两种设置中散射矩阵相同的原因。
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