We discuss representation of certain functions of the Laplace operator $Δ$ as Dirichlet-to-Neumann maps for appropriate elliptic operators in half-space. A classical result identifies $(-Δ)^{1/2}$, the square root of the $d$-dimensional Laplace operator, with the Dirichlet-to-Neumann map for the $(d + 1)$-dimensional Laplace operator $Δ_{t,x}$ in $(0, \infty) \times \mathbf{R}^d$. Caffarelli and Silvestre extended this to fractional powers $(-Δ)^{α/2}$, which correspond to operators $\nabla_{t,x} (t^{1 - α} \nabla_{t,x})$. We provide an analogous result for all complete Bernstein functions of $-Δ$ using Krein's spectral theory of strings.
Two sample applications are provided: a Courant--Hilbert nodal line theorem for harmonic extensions of the eigenfunctions of non-local Schrödinger operators $ψ(-Δ) + V(x)$, as well as an upper bound for the eigenvalues of these operators. Here $ψ$ is a complete Bernstein function and $V$ is a confining potential.
Let $φ_λ$ be an eigenfunction of the Laplace-Beltrami operator on a smooth compact Riemannian manifold $(M,g)$, i.e., $Δ_g φ_λ + λφ_λ=0$. We show that $φ_λ$ satisfies a local Bernstein inequality, namely for any geodesic ball $B_g(x,r)$ in $M$ there holds: $\sup_{B_g(x,r)}|\nablaφ_λ|\leq C_δ\max\left\{\frac{\sqrtλ\log^{2+δ}λ}{r},λ\log^{2+δ}λ\right\}\sup_{B_g(x,r)}|φ_λ|$. We also prove analogous inequalities for solutions of elliptic PDEs in terms of the frequency function.
By observing that the fractional Caputo derivative can be expressed in terms of a multiplicative convolution operator, we introduce and study a class of such operators which also have the same self-similarity property as the Caputo derivative. We proceed by identifying a subclass which is in bijection with the set of Bernstein functions and we provide several representations of their eigenfunctions, expressed in terms of the corresponding Bernstein function, that generalize the Mittag-Leffler function. Each eigenfunction turns out to be the Laplace transform of the right-inverse of a non-decre…
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…
In the present paper, we solve three boundary value problems related to the temperature field in oil strata -- the fractional extensions of the incomplete lumped formulation and lumped formulation in the linear case and the fractional generalization of the incomplete lumped formulation in the radial case. By using the Caputo differintegral operator and the Laplace transform, the solutions are obtained in integral forms where the integrand is expressed in terms of the convolution of some auxiliary functions of Wright function type. A generalization of the Laplace transform convolution theorem, …
We prove a Liouville-type theorem that is one-dimensional symmetry and classification results for non-negative $L^q$-viscosity solutions of the equation \begin{equation*} -\mathcal{M}_{λ, Λ}^{\pm}(D^2u)\pm |Du|^p=0, x\in \mathbb{R}_+^n, \end{equation*} with boundary condition $u(\tilde{x},0)=M\geq 0, \tilde{x}\in \mathbb{R}^{n-1}$, where $\mathcal{M}_{λ, Λ}^{\pm}$ are the Pucci's operators with parameters $λ, Λ\in \mathbb{R}_+$ $0<λ\leq Λ$ and $p>1$. The results are an extension of the results by Porreta and Verón in arXiv:0805.2533 for the case $p\in (1,2]$ and by o Filippucci, Pucci and Soup…
This article reviews several definitions of the fractional Laplace operator (-Delta)^{alpha/2} (0 < alpha < 2) in R^d, also known as the Riesz fractional derivative operator, as an operator on Lebesgue spaces L^p, on the space C_0 of continuous functions vanishing at infinity and on the space C_{bu} of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner's subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces …
In this paper we apply estimates of the norms of Sobolev extension operators to the spectral estimates of of the first nontrivial Neumann eigenvalue of the Laplace operator in non-convex extension domains. As a consequence we obtain a connection between resonant frequencies of free membranes and the smallest-circle problem (initially proposed by J.~J.~Sylvester in 1857).
Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学）；Department of Mathematics, Purdue University（普渡大学）
It has recently been shown that complete Bernstein functions of the Laplace operator map the Dirichlet boundary condition of a related elliptic PDE to the Neumann boundary condition. The importance of this mapping consists in being able to convert problems involving non-local operators, like fractional Laplacians, into ones that only involve differential operators. We generalise this result to diffusion operators associated with stochastic differential equations, using a method which is entirely based on stochastic analysis.
It is shown that if $p \ge 3$ and $u \in W^{1,p}(Ω,\mathbb{R}^N)$ solves the inhomogenous $p$-Laplace system \[ \operatorname{div} (|\nabla u|^{p-2} \nabla u) = f, \qquad f \in W^{1,p'}(Ω,\mathbb{R}^N), \] then locally the gradient $\nabla u$ lies in the fractional Nikol'skii space $\mathcal{N}^{θ,2/θ}$ with any $θ\in [ \tfrac{2}{p}, \tfrac{2}{p-1} )$. To the author's knowledge, this result is new even in the case of $p$-harmonic functions, slightly improving known $\mathcal{N}^{2/p,p}$ estimates. The method used here is an extension of the one used by A. Cellina in the case $2 \le p < 3$ to s…
We identify a class of non-local integro-differential operators $K$ in $\mathbb{R}$ with Dirichlet-to-Neumann maps in the half-plane $\mathbb{R} \times (0, \infty)$ for appropriate elliptic operators $L$. More precisely, we prove a bijective correspondence between Lévy operators $K$ with non-local kernels of the form $ν(y - x)$, where $ν(x)$ and $ν(-x)$ are completely monotone functions on $(0, \infty)$, and elliptic operators $L = a(y) \partial_{xx} + 2 b(y) \partial_{x y} + \partial_{yy}$. This extends a number of previous results in the area, where symmetric operators have been studied: the…
We introduce three representation formulas for the fractional $p$-Laplace operator in the whole range of parameters $0<s<1$ and $1<p<\infty$. Note that for $p\ne 2$ this a nonlinear operator. The first representation is based on a splitting procedure that combines a renormalized nonlinearity with the linear heat semigroup. The second adapts the nonlinearity to the Caffarelli-Silvestre linear extension technique. The third one is the corresponding nonlinear version of the Balakrishnan formula. We also discuss the correct choice of the constant of the fractional $p$-Laplace operator in order to …
The technique of Caffarelli and Silvestre, characterizing the fractional Laplacian as the Dirichlet-to-Neumann map for a function U satisfying an elliptic equation in the upper half space with one extra spatial dimension, is shown to hold for general positive, non-integer orders of the fractional Laplace operator, by showing an equivalence between the H^s norm on the boundary and a suitable higher-order seminorm of U.
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 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 …
