Caffarelli-Silvestre Extension Research Articles

Fractional Sobolev spaces on Riemannian manifolds

AbstractThis article studies the canonical Hilbert energy $$H^{s/2}(M)$$ H s / 2 ( M ) on a Riemannian manifold for $$s\in (0,2)$$ s ∈ ( 0 , 2 ) , with particular focus on the case of closed manifolds. Several equivalent definitions for this energy and the fractional Laplacian on a manifold are given, and they are shown to be identical up to explicit multiplicative constants. Moreover, the precise behavior of the kernel associated with the singular integral definition of the fractional Laplacian is obtained through an in-depth study of the heat kernel on a Riemannian manifold. Furthermore, a monotonicity formula for stationary points of functionals of the type $${\mathcal {E}}(v)=[v]^2_{H^{s/2}(M)}+\int _M F(v) \, dV$$ E ( v ) = [ v ] H s / 2 ( M ) 2 + ∫ M F ( v ) d V , with $$F\ge 0$$ F ≥ 0 , is given, which includes in particular the case of nonlocal s-minimal surfaces. Finally, we prove some estimates for the Caffarelli–Silvestre extension problem, which are of general interest. This work is motivated by Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023), which defines nonlocal minimal surfaces on closed Riemannian manifolds and shows the existence of infinitely many of them for any metric on the manifold, ultimately proving the nonlocal version of a conjecture of Yau (Ann Math Stud 102:669–706, 1982). Indeed, the definitions and results in the present work serve as an essential technical toolbox for the results in Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023).

Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds

This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.

Fractal dimension of potential singular points set in the Navier–Stokes equations under supercritical regularity

The main objective of this paper is to answer the questions posed by Robinson and Sadowski [22, p. 505, Commun. Math. Phys., 2010] for the Navier–Stokes equations. Firstly, we prove that the upper box dimension of the potential singular points set $\mathcal {S}$ of suitable weak solution $u$ belonging to $L^{q}(0,T;L^{p}(\mathbb {R}^{3}))$ for $1\leq \frac {2}{q}+\frac {3}{p}\leq \frac 32$ with $2\leq q<\infty$ and $2< p<\infty$ is at most $\max \{p,q\}(\frac {2}{q}+\frac {3}{p}-1)$ in this system. Secondly, it is shown that $1-2s$ dimension Hausdorff measure of potential singular points set of suitable weak solutions satisfying $u\in L^{2}(0,T;\dot {H}^{s+1}(\mathbb {R}^{3}))$ for $0\leq s\leq \frac 12$ is zero, whose proof relies on Caffarelli–Silvestre's extension. Inspired by Barker–Wang's recent work [1], this further allows us to discuss the Hausdorff dimension of potential singular points set of suitable weak solutions if the gradient of the velocity is under some supercritical regularity.

Landis-type conjecture for the half-Laplacian

In this paper, we study the Landis-type conjecture, i.e., unique continuation property from infinity, of the fractional Schrödinger equation with drift and potential terms. We show that if any solution of the equation decays at a certain exponential rate, then it must be trivial. The main ingredients of our proof are the Caffarelli-Silvestre extension and Armitage’s Liouville-type theorem.

The fractional p,-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems

This article investigates nonlocal, quasilinear generalizations of the classical biharmonic operator (-Delta )^2. These fractional p -biharmonic operators appear naturally in the variational characterization of the optimal fractional Poincaré constants in Bessel potential spaces. We study the following basic questions for anisotropic fractional p -biharmonic systems: existence and uniqueness of weak solutions to the associated interior source and exterior value problems, unique continuation properties, monotonicity relations, and inverse problems for the exterior Dirichlet-to-Neumann maps. Furthermore, we show the UCP for the fractional Laplacian in all Bessel potential spaces H^{t,p} for any tin {mathbb R}, 1 le p < infty and s in {mathbb R}_+ {setminus } {mathbb N}: If uin H^{t,p}({mathbb R}^n) satisfies (-Delta )^su=u=0 in a nonempty open set V, then uequiv 0 in {mathbb R}^n. This property of the fractional Laplacian is then used to obtain a UCP for the fractional p -biharmonic systems and plays a central role in the analysis of the associated inverse problems. Our proofs use variational methods and the Caffarelli–Silvestre extension.

Singularities of fractional Emden's equations via Caffarelli-Silvestre extension

We study the isolated singularities of functions satisfying(E)(−Δ)sv±|v|p−1v=0inΩ∖{0},v=0inRN∖Ω, where 0<s<1, p>1 and Ω is a bounded domain containing the origin. We use the Caffarelli-Silvestre extension to R+×RN. We emphasize the obtention of a priori estimates and analyse the set of self-similar solutions via energy methods to characterize the singularities.

New Brezis–Van Schaftingen–Yung–Sobolev type inequalities connected with maximal inequalities and one parameter families of operators

Motivated by the recent characterization of Sobolev spaces due to Brezis–Van Schaftingen–Yung we prove new weak-type inequalities for one parameter families of operators connected with mixed norm inequalities. The novelty of our approach comes from the fact that the underlying measure space incorporates the parameter as a variable. We also show that our framework can be adapted to treat related characterizations of Sobolev spaces obtained earlier by Bourgain–Nguyen. The connection to classical and fractional order Sobolev spaces is shown through the use of generalized Riesz potential spaces and the Caffarelli–Silvestre extension principle. Higher order inequalities are also considered. We indicate many examples and applications to PDE's and different areas of Analysis, suggesting a vast potential for future research. In a different direction, and inspired by methods originally due to Gagliardo and Garsia, we obtain new maximal inequalities which combined with mixed norm inequalities are applied to obtain Brezis–Van Schaftingen–Yung type inequalities in the context of Calderón–Campanato spaces. In particular, Log versions of the Gagliardo–Brezis–Van Schaftingen–Yung spaces are introduced and compared with corresponding limiting versions of Calderón–Campanato spaces, resulting in a sharpening of recent inequalities due to Crippa–De Lellis and Brué–Nguyen.

Infinitely many solutions for fractional elliptic systems involving critical nonlinearities and Hardy potentials

This paper is dedicated to studying the existence of multiple nontrivial solutions for a class of singular fractional elliptic systems involving critical nonlinearities and Hardy potentials in RN. Based upon the Caffarelli–Silvestre extension method and the Krasnoselskii genus theory, together with the symmetric criticality principle of Palais, we establish several existence results of infinitely many solutions under certain appropriate hypotheses on the weights and the parameters.

Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy

AbstractTime-fractional partial differential equations are nonlocal-in-time and show an innate memory effect. Previously, examples like the time-fractional Cahn-Hilliard and Fokker-Planck equations have been studied. In this work, we propose a general framework of time-fractional gradient flows and we provide a rigorous analysis of well-posedness using the Faedo-Galerkin approach. Furthermore, we investigate the monotonicity of the energy functional of time-fractional gradient flows. Interestingly, it is still an open problem whether the energy is dissipating in time. This property is essential for integer-order gradient flows and many numerical schemes exploit this steepest descent characterization. We propose an augmented energy functional, which includes the history of the solution. Based on this new energy, we prove the equivalence of a time-fractional gradient flow to an integer-order one. This correlation guarantees the dissipating character of the augmented energy. The state function of the integer-order gradient flow acts on an extended domain similar to the Caffarelli-Silvestre extension for the fractional Laplacian. Additionally, we present a numerical scheme for solving time-fractional gradient flows, which is based on kernel compressing methods and reduces the problem to a system of ordinary differential equations. We illustrate the behavior of the original and augmented energy in the case of the Ginzburg-Landau energy.

Deep Ritz Method for the Spectral Fractional Laplacian Equation Using the Caffarelli--Silvestre Extension

In this paper, we propose a novel method for solving high-dimensional spectral fractional Laplacian equations. Using the Caffarelli-Silvestre extension, the $d$-dimensional spectral fractional equation is reformulated as a regular partial differential equation of dimension $d+1$. We transform the extended equation as a minimal Ritz energy functional problem and search for its minimizer in a special class of deep neural networks. Moreover, based on the approximation property of networks, we establish estimates on the error made by the deep Ritz method. Numerical results are reported to demonstrate the effectiveness of the proposed method for solving fractional Laplacian equations up to ten dimensions. Technically, in this method, we design a special network-based structure to adapt to the singularity and exponential decaying of the true solution. Also, A hybrid integration technique combining Monte Carlo method and sinc quadrature is developed to compute the loss function with higher accuracy.

Fractional Besov Trace/Extension-Type Inequalities via the Caffarelli–Silvestre Extension

Let $$u(\cdot ,\cdot )$$ be the Caffarelli–Silvestre extension of f. The first goal of this article is to establish the fractional trace-type inequalities involving the Caffarelli–Silvestre extension $$u(\cdot ,\cdot )$$ of f. In doing so, firstly, we establish the fractional Sobolev/logarithmic Sobolev/Hardy trace inequalities in terms of $$\nabla _{(x,t)}u(x,t).$$ Then, we prove the fractional anisotropic Sobolev/logarithmic Sobolev/Hardy trace inequalities in terms of $$ {\partial _{t} u(x,t)}$$ or $$(-\varDelta )^{-\gamma /2}u(x,t)$$ only. Moreover, based on an estimate of the Fourier transform of the Caffarelli–Silvestre extension kernel and the sharp affine weighted $$L^p$$ Sobolev inequality, we prove that the $${\dot{H}}^{-\beta /2}({\mathbb {R}}^n)$$ norm of f(x) can be controlled by the product of the weighted $$L^p$$ -affine energy and the weighted $$L^p$$ -norm of $${\partial _{t} u(x,t)}.$$ The second goal of this article is to characterize non-negative measures $$\mu $$ on $${\mathbb {R}}^{n+1}_+$$ such that the embeddings $$\begin{aligned} \Vert u(\cdot ,\cdot )\Vert _{L^{q_0,p_0}_{\mu }({\mathbb {R}}^{n+1})}\lesssim \Vert f\Vert _{{\dot{\varLambda }}^{p,q}_\beta ({\mathbb {R}}^n)} \end{aligned}$$ hold for some $$p_0$$ and $$q_0$$ depending on p and q which are classified in three different cases: (1) $$p=q\in (n/(n+\beta ),1];$$ (2) $$(p,q)\in (1,n/\beta )\times (1,\infty );$$ (3) $$(p,q)\in (1,n/\beta )\times \{\infty \}.$$ For case (1), the embeddings can be characterized in terms of an analytic condition of the variational capacity minimizing function, the iso-capacitary inequality of open balls, and other weak-type inequalities. For cases (2) and (3), the embeddings are characterized by the iso-capacitary inequality for fractional Besov capacity of open sets.

Spectral, Tensor and Domain Decomposition Methods for Fractional PDEs

Abstract Fractional PDEs have recently found several geophysics and imaging science applications due to their nonlocal nature and their flexibility in capturing sharp transitions across interfaces. However, this nonlocality makes it challenging to design efficient solvers for such problems. In this paper, we introduce a spectral method based on an ultraspherical polynomial discretization of the Caffarelli–Silvestre extension to solve such PDEs on rectangular and disk domains. We solve the discretized problem using tensor equation solvers and thus can solve higher-dimensional PDEs. In addition, we introduce both serial and parallel domain decomposition solvers. We demonstrate the numerical performance of our methods on a 3D fractional elliptic PDE on a cube as well as an application to optimization problems with fractional PDE constraints.

Embeddings of function spaces via the Caffarelli–Silvestre extension, capacities and Wolff potentials

Let Pαf(x,t) be the Caffarelli–Silvestre extension of a smooth function f(x):Rn→R+n+1≔Rn×(0,∞). The purpose of this article is twofold. Firstly, we want to characterize a nonnegative measure μ on R+n+1 such that f(x)→Pαf(x,t) induces bounded embeddings from the Lebesgue spaces Lp(Rn) to the Lq(R+n+1,μ). On one hand, these embeddings will be characterized by using a newly introduced Lp−capacity associated with the Caffarelli–Silvestre extension. In doing so, the mixed norm estimates of Pαf(x,t), the dual form of the Lp−capacity, the Lp−capacity of general balls, and a capacitary strong type inequality will be established, respectively. On the other hand, when p>q>1, these embeddings will also be characterized in terms of the Hedberg–Wolff potential of μ. Secondly, we characterize a nonnegative measure μ on R+n+1 such that f(x)→Pαf(x,t) induces bounded embeddings from the homogeneous Sobolev spaces Ẇβ,p(Rn) to the Lq(R+n+1,μ) in terms of the fractional perimeter of open sets for endpoint cases and the fractional capacity for general cases.

The Lewy-Stampacchia inequality for the fractional Laplacian and its application to anomalous unidirectional diffusion equations

&lt;p style='text-indent:20px;'&gt;In this paper, we consider a Lewy-Stampacchia-type inequality for the fractional Laplacian on a bounded domain in Euclidean space. Using this inequality, we can show the well-posedness of fractional-type anomalous unidirectional diffusion equations. This study is an extension of the work by Akagi-Kimura (2019) for the standard Laplacian. However, there exist several difficulties due to the nonlocal feature of the fractional Laplacian. We overcome those difficulties employing the Caffarelli-Silvestre extension of the fractional Laplacian.&lt;/p&gt;

Uniqueness for linear integro-differential equations in the real line and applications

In this work we prove the uniqueness of solutions to the nonlocal linear equation L varphi - c(x)varphi = 0 in mathbb {R}, where L is an elliptic integro-differential operator, in the presence of a positive solution or of an odd solution vanishing only at zero. As an application, we deduce the nondegeneracy of layer solutions (bounded and monotone solutions) to the semilinear problem L u = f(u) in mathbb {R} when the nonlinearity is of Allen–Cahn type. To our knowledge, this is the first work where such uniqueness and nondegeneracy results are proven in the nonlocal framework when the Caffarelli–Silvestre extension technique is not available. Our proofs are based on a nonlocal Liouville-type method developed by Hamel, Ros-Oton, Sire, and Valdinoci for nonlinear problems in dimension two.

Almost minimizers for certain fractional variational problems

A notion of almost minimizers is introduced for certain variational problems governed by the fractional Laplacian, with the help of the Caffarelli–Silvestre extension. In particular, almost fractional harmonic functions and almost minimizers for the fractional obstacle problem with zero obstacle are treated. It is shown that for a certain range of parameters, almost minimizers are almost Lipschitz or C 1 , β C^{1,\beta } -regular.

Global existence and blowup of solutions to semilinear fractional reaction-diffusion equation with singular potential

We consider the following semilinear fractional reaction-diffusion equation with singular potential{Asu=−ut|x|2s+|u|p−2u,(x,t)∈Ω×(0,∞),u=0,(x,t)∈(RN∖Ω)×[0,∞),u(x,0)=u0,x∈Ω. Here As(0<s<1) represents spectral fractional Laplacian operator, 2<p<2s⁎, 2s⁎=2NN−2s is critical exponent of fractional Sobolev trace embedding inequality, N>2s, and Ω is bounded domain in RN with smooth boundary. We first prove that the solution exists globally under appropriate hypotheses. And due to the non-locality of fractional Laplacian operator, we use the Caffarelli-Silvestre extension method to convert the non-local problem into a variable local problem. Being based upon this, by means of potential well, we obtain decay estimate and long time asymptotic behavior of global solution, as well as blow-up behavior of local solution.

The finite element method for fractional diffusion with spectral fractional Laplacian

This paper focuses on the finite element method for Caputo‐type parabolic equation with spectral fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0,1) and the spatial derivative is the spectral fractional Laplacian. The time discretization is based on the Hadamard finite‐part integral (or the finite‐part integral in the sense of Hadamard), where the piecewise linear interpolation polynomials are used. The spatial fractional Laplacian is lifted to the local spacial derivative by using the Caffarelli–Silvestre extension, where the finite element method is used. Full‐discretization scheme is constructed. The convergence and error estimates are obtained. Finally, numerical experiments are presented which support the theoretical results.

Hp-FEM for the fractional heat equation

Abstract We consider a time-dependent problem generated by a nonlocal operator in space. Applying for the spatial discretization a scheme based on $hp$-finite elements and a Caffarelli–Silvestre extension we obtain a semidiscrete semigroup. The discretization in time is carried out by using $hp$-discontinuous Galerkin based time stepping. We prove exponential convergence for such a method in an abstract framework for the discretization in the spatial domain $\varOmega $.

A quantitative stability estimate for the fractional Faber-Krahn inequality

We prove a quantitative version of the Faber-Krahn inequality for the first eigenvalue of the fractional Dirichlet-Laplacian of order s. This is done by using the so-called Caffarelli-Silvestre extension and adapting to the nonlocal setting a trick by Hansen and Nadirashvili. The relevant stability estimate comes with an explicit constant, which is stable as the fractional order of differentiability goes to 1.