arXiv research

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.\$

In the first part of the paper we survey some nonlocal flows of convex plane curves ever studied so far and discuss properties of the flows related to enclosed area and length, especially the isoperimetric ratio and the isoperimetric difference. We also study a new nonlocal flow of convex plane curves and discuss its e…

Motivated by Pan-Yang [PY] and Ma-Cheng [MC], we study a general linear nonlocal curvature flow for convex closed plane curves and discuss the short time existence and asymptotic convergence behavior of the flow. Due to the linear structure of the flow, this partial differential equation problem can be resolved using a…

Estimates for VPMCF show ancient MCF solutions and finite-time behavior.

Study curves evolving on hypersurfaces with free boundaries, preserving length.

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

Study on dynamic curves with elastic energy and spontaneous curvature.

Smoothness of graphs evolving by fractional mean curvature is proven.

Flow deforms locally convex curves to curves of constant k-order width.

The Sinkhorn flow is a gradient flow in a nonlocal Wasserstein geometry.

Solves curve migration problem with elastic flows.

We study the geometric flow of a planar curve driven by its curvature and the normal derivative of its capacity potential. Under a convexity condition that is natural to our problem, we establish long term existence and large time asymptotics of this flow.

New method extracts stochastic laws from data, including Lévy noise.

In this paper, we establish that: Suppose a closed Riemannian manifold $(M^n,g_0)$ of dimension $\geq 8$ is not locally conformally flat, then the Paneitz-Sobolev constant of $M^n$ has the property that $q(g_0)<q(S^n)$. The analogy of this result was obtained by T. Aubin in 1976 and had been used to solve the Yamabe pr…

Flow approach solves Toda system equations.

Two flows for convex curves converge to circles smoothly.

We study the phase field method for the volume preserving mean curvature flow. Given an initial $C^1$ hypersurface we proved the existence of the weak solution for the volume preserving mean curvature flow via the reaction diffusion equation with a nonlocal term. We also show the monotonicity formula and the density up…

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…

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

Study on evolving interfaces with complex curvature and density effects.

Approximates nonlocal curvature of curves using splines.

Paper connects Bäcklund transformations to nonlocal pseudosymmetries.

The paper proves removable singularity for nonlocal minimal graphs.

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

Flow deforms locally convex curves into target curves.

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

Proposes a new nonlocal curvature tensor concept.

The paper proves an inequality and describes a curve flow in centro-affine geometry.

New theorem for nonlocal minimal surfaces in any dimension.

We study a class of nonlocal, energy-driven dynamical models that govern the motion of closed, embedded curves from both an energetic and dynamical perspective. Our energetic results provide a variety of ways to understand physically motivated energetic models in terms of more classical, combinatorial measures of compl…

In this paper we consider the evolution of sets by a fractional mean curvature flow. Our main result states that for any dimension $n > 2$, there exists an embedded surface in $\mathbb R^n$ evolving by fractional mean curvature flow, which developes a singularity before it can shrink to a point. When $n > 3$ this resul…

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.

Study of nonlocal mean curvature and symmetry of surfaces.

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

We introduce a fractional Yamabe flow involving nonlocal conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds, and show that on the conformal spheres $(\Sn, [g_{\Sn}])$, it converges to the standard sphere up to a Möbius diffeomorphism. This result allows us to obtain extinct…

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.

MetaNOR learns common nonlocal kernels for efficient metamaterial modeling.

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…

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…

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…

A new image interpolation model using sparse representation and nonlocal linear regression.

A fundamental problem in Bayesian inference and statistical machine learning is to efficiently sample from multimodal distributions. Due to metastability, multimodal distributions are difficult to sample using standard Markov chain Monte Carlo methods. We propose a new sampling algorithm based on a birth-death mechanis…

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.

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

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