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 $\C$.
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 $\mathbb{S}^N$.
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.
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…
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.
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.
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}.
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 …
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.
Under integral restrictions on dilatations, it is proved existence theorems for the degenerate Beltrami equations with two characteristics and, in particular, to the Beltrami equations of the second type that play a great role in many problems of mathematical physics and to the so--called reduced Beltrami equations that also have significant applications.
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.
It is shown that many recent and new results on the existence of ACL homeomorphic (and more strong) solutions for the Beltrami equations with integral constraints follow from our extension of the well--known Lehto existence theorem.
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.
We introduce an iterative method to prove the existence and uniqueness of the complex-valued nonlinear elliptic PDE of the form $ -Δu + F(u) = f $ with Dirichlet or Neumann boundary conditions on a precompact domain $ Ω\subset \mathbb{R}^{n}$, where $ F : \mathbb{C} \rightarrow \mathbb{C} $ is Lipschitz. The same method gives a solution to $ - Δ_{g} u + F(u) = f $ for these boundary conditions on a smooth, compact Riemannian manifold $ (M, g) $ with $ \mathcal{C}^{1} $ boundary, where $ - Δ_{g} $ is the Laplace-Beltrami operator. We also apply parametrix methods to discuss an integral version …
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…
An essentially unique homeomorphic solution to the Beltrami equation was found in the 1960s using the theory of Calderón-Zygmund and singular integral operators in $L^p(\mathbb C)$. We will present an alternative method to solve the Beltrami equation using the Hodge star operator and standard elliptic PDE theory. We will also discuss a different method to prove the regularity of the solution. This approach is partially based on work by Dittmar.
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…
