arXiv research

New theorem for nonlocal minimal surfaces in any dimension.

New nonlocal minimal surfaces on manifolds, proving Yau's conjecture.

We prove the existence of the analog of Lawson's minimal cones for a notion of nonlocal minimal surface introduced by Caffarelli, Roquejoffre and Savin, and establish their stability/instability in low dimensions. In particular we find that there are nonlocal stable minimal cones in dimension 7, in contrast with the ca…

Study nonlocal minimal surfaces for minimal surfaces in 3D manifolds.

Study fractional Allen-Cahn equation and nonlocal minimal surfaces, improving energy and perimeter estimates.

New mathematical surfaces without boundaries found.

The nonlocal $s$-fractional minimal surface equation for $Σ= \partial E$ where $E$ is an open set in $R^N$ is given by $H_Σ^ s (p) := \int_{R^N} \frac {χ_E(x) - χ_{E^c}(x)} {|x-p|^{N+s}}\, dx \ =\ 0 \quad \text{for all } p\in Σ.$ Here $0<s<1$, $χ$ designates characteristic function, and the integral is understood i…

An infinite sequence of commuting nonpolynomial contact symmetries of the two-dimensional minimal surface equation is constructed. Local and nonlocal conservation laws for $n$-dimensional minimal area surface equation are obtained by using the Noether identity.

This paper develops a new nonlocal approximation method for minimal surfaces, proving robust estimates and separation properties.

We consider the class of measurable functions defined in all of $\mathbb{R}^n$ that give rise to a nonlocal minimal graph over a ball of $\mathbb{R}^n$. We establish that the gradient of any such function is bounded in the interior of the ball by a power of its oscillation. This estimate, together with previously known…

The paper proves removable singularity for nonlocal minimal graphs.

Investigates convexity of minimizers under mass constraint using nonlocal perimeter and potential.

The paper studies nonlocal isoperimetric problems in hyperbolic space and finds unique minimizers for small volumes.

For surfaces without boundary, nonlocal notions of directional and mean curvatures have been recently given. Here, we develop alternative notions, special cases of which apply to surfaces with boundary. Our main tool is a new fractional or nonlocal area functional for compact surfaces.

Study on capillarity minimizers with nonlocal repulsion and gravity, proving existence and nonexistence.

Study of nonlocal mean curvature and symmetry of surfaces.

Study finds conditions for global minimizers on curved manifolds with fast diffusion and nonlocal interactions.

We are concerned with hypersurfaces of $\mathbb{R}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing sph…

The notion of Nonlocal Mean Curvature (NMC) appears recently in the mathematics literature. It is an extrinsic geometric quantity that is invariant under global reparameterization of a surface and provide a natural extension of the classical mean curvature. We describe some properties of the NMC and the quasilinear dif…

The paper proves a Fenchel theorem for Gauss maps and shows circles and disks minimize certain energies.

Here a new notion of fractional length of a smooth curve, which depends on a parameter $σ$, is introduced that is analogous to the fractional perimeter functional of sets that has been studied in recent years. It is shown that in an appropriate limit the fractional length converges to the traditional notion of length u…

Nonlocal neural networks have been proposed and shown to be effective in several computer vision tasks, where the nonlocal operations can directly capture long-range dependencies in the feature space. In this paper, we study the nature of diffusion and damping effect of nonlocal networks by doing spectrum analysis on t…

Approximates nonlocal curvature of curves using splines.

Paper connects Bäcklund transformations to nonlocal pseudosymmetries.

We present infinitely many nonlocal conservation laws, a pair of compatible local Hamiltonian structures and a recursion operator for the equations describing surfaces in three-dimensional space that admit nontrivial deformations which preserve both principal directions and principal curvatures (or, equivalently, the s…

The paper proves a nonlocal version of the Alexandrov Theorem for smooth boundaries.

Proposes a new nonlocal curvature tensor concept.

New geometric insights reveal properties of adversarial training problems.

Study introduces fractional mass concept for surfaces, proving its convergence.

Study shows global invertibility in nonlinear elasticity with vanishing self-repulsion term.

The paper characterizes stochastic completeness on Riemannian manifolds using nonlocal conditions.

Nonlocal Bayesian modeling for continuous spatio-temporal dynamics

Researchers solve a nonlocal parabolic equation on manifolds using source-to-solution maps.

Study proposes a nonlocal approximation of the Willmore functional using fractional Allen-Cahn energies.

Paper proves existence and uniqueness of solutions to nonlocal systems, generalizing stochastic game theory.

MetaNOR learns common nonlocal kernels for efficient metamaterial modeling.

We study hypersurfaces of $\mathbb{R}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We establish the existence of a smooth branch of periodic cylinders in $\mathbb{R}^N$, $N\geq 2$, all of th…

Survey discusses recent advances on Brakke flows and their properties.

In the present work, the integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type are effectively constructed. For achieving this aim, first of all, the problem on the canonical form of a special type for compatible nonlocal Poisson brackets of hydrodynamic type is solv…

The study proves inequalities for complex operators on curved spaces.

Rediscovered by a systematic search, a forgotten class of integrable surfaces is shown to disprove the Finkel-Wu conjecture. The associated integrable nonlinear partial differential equation $z_{yy} + (1/z)_{xx} + 2 = 0$ possesses a zero curvature representation, a third-order symmetry, and a nonlocal transformatio…

For any $α>0,$ we study $k^α$-type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in $C^{\infty}$-norm.$\$Other relevant $k^α$-type nonlocal flow is also discussed when $α\geq1.\$

Proves uniqueness of solutions for a nonlocal Liouville equation with finite Q-curvature.

In this paper, we study a nonlocal elliptic problem with the fractional Laplacian on $R^n$. We show that the problem has infinite positive solutions in $C^τ(R^n)\bigcap H^α_{loc}(R^n)$. Moreover each of these solutions tends to some positive constant limit at infinity. We extend Lin's result to the nonlocal problem on …

The paper proves well-posedness of nonlocal PDEs related to stochastic control problems.

Develops a nonlocal PINN framework using PDDO for better solution of PDEs with sharp gradients.

Smoothness of graphs evolving by fractional mean curvature is proven.

First variation of fractional $k$-dimensional measure for submanifolds