Results 71 to 80 of about 101,180 (174)
An extension problem for the CR fractional Laplacian [PDF]
Advances in Mathematics, 2015 We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre.
Frank, Rupert L. +3 more
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On the Laplacian and fractional Laplacian in an exterior domain
Advances in Differential Equations, 2012 We see that the generalized Fourier transform due to A.G. Ramm for the case of $n=3$ space dimensions remains valid, with some modifications, for all space dimensions $n\ge 2$. We use the resulting spectral representation of the exterior Laplacian to study exterior problems. In particular the Fourier splitting method developed by M.E.
Kosloff, Leonardo, Schonbek, Tomas
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Uniqueness of Radial Solutions for the Fractional Laplacian [PDF]
Communications on Pure and Applied Mathematics, 2015 AbstractWe prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (−Δ)s with s ∊ (0,1) for any space dimensions N ≥ 1. By extending a monotonicity formula found by Cabré and Sire , we show that the linear equation urn:x-wiley:00103640:media:cpa21591:cpa21591-math-0001 has at most one
Frank, Rupert L. +2 more
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The second eigenvalue of the fractional p-Laplacian [PDF]
Advances in Calculus of Variations, 2016 AbstractWe consider the eigenvalue problem for the fractional p-Laplacian in an open bounded, possibly disconnected set ${\Omega\subset\mathbb{R}^{n}}$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfunctions, we show that the second eigenvalue ${\lambda_{2}(\Omega)}$ is well-defined, and we ...
BRASCO, Lorenzo, Parini, Enea
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On fractional powers of singular perturbations of the Laplacian [PDF]
Journal of Functional Analysis, 2018 We qualify a relevant range of fractional powers of the so-called Hamiltonian of point interaction in three dimensions, namely the singular perturbation of the negative Laplacian with a contact interaction supported at the origin. In particular we provide an explicit control of the domain of such a fractional operator and of its decomposition into ...
Vladimir Georgiev +3 more
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Eigenvalue estimates for the fractional Laplacian on lattice subgraphs [PDF]
arXiv, 2023 We introduce the the fractional Laplacian on a subgraph of a graph with Dirichlet boundary condition. For a lattice graph, we prove the upper and lower estimates for the sum of the first $k$ Dirichlet eigenvalues of the fractional Laplacian, extending the classical results by Li-Yau and Kr\"{o}ger.
arxiv
The fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation [PDF]
arXiv, 2023 In this paper we construct the fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation approach. In addition, we characterize the powers of the fractional sub-Laplacian in the Heisenberg group, and as a byproduct we compute the $k$-th order momenta with respect to the heat kernel.
arxiv
Fractional Laplacians and extension problems: the higher rank case [PDF]
arXiv, 2016 The aim of this paper is two-fold: first, we look at the fractional Laplacian and the conformal fractional Laplacian from the general framework of representation theory on symmetric spaces and, second, we construct new boundary operators with good conformal properties that generalize the fractional Laplacian using an extension problem in which the ...
arxiv
Eigenvalues for systems of fractional $p$-Laplacians [PDF]
Rocky Mountain Journal of Mathematics, 2018 We study the eigenvalue problem for a system of fractional $p-$Laplacians, that is, $$ \begin{cases} (- _p)^r u = \dfrac p|u|^{ -2}u|v|^ &\text{in } ,\vspace{.1cm} (- _p)^s u = \dfrac p|u|^ |v|^{ -2}v &\text{in } , u=v=0 &\text{in } ^c=\R^N\setminus . \end{cases} $$ We show that there is a first (smallest) eigenvalue that
Pezzo, Leandro M. Del, Rossi, Julio D.
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Nontrivial Solution for the Fractional p-Laplacian Equations via Perturbation Methods
Advances in Mathematical Physics, 2017 We study the existence of nontrivial solution of the following equation without compactness: (-Δ)pαu+up-2u=f(x,u), x∈RN, where N,p≥2, α∈(0,1), (-Δ)pα is the fractional p-Laplacian, and the subcritical p-superlinear term f∈C(RN×R) is 1-periodic in xi ...
Huxiao Luo, Shengjun Li, Xianhua Tang
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