arXiv research

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

Study on dynamic curves with elastic energy and spontaneous curvature.

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

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

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

Study on evolving interfaces with complex curvature and density effects.

Approximates nonlocal curvature of curves using splines.

Proposes a new nonlocal curvature tensor concept.

Flow deforms locally convex curves into target curves.

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

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

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

Study of nonlocal mean curvature and symmetry of surfaces.

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…

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

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.

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

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…

The study proves inequalities for complex operators on curved spaces.

Study on existence of ground states for free energy on hyperbolic space.

Smoothness of graphs evolving by fractional mean curvature is proven.

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

Study on existence of ground states on curved spaces with conditions on potential growth.

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

The study proves that sets with constant nonlocal curvature are composed of equal balls under certain conditions.

We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controls its $C^2$-distance fro…

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

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…

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

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…

Study optimal partitions on spheres using fractional Q-curvature and variational methods.

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…

We solve the problem of reducing to the simplest and convenient for our purposes, canonical form for an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature in order to get an effective construction of the integrable hierarchies related to all…

Study on consensus formation in manifolds with curvature constraints.

Magnitude study on manifolds using fractional Laplacian.

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…

New method models dewetting of anisotropic particles using numerical techniques.

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…

The nonlinear equations for the general nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived and the integrability of these equations by the method of inverse scattering problem is proved. For these equations, the Lax pairs with a spectral parameter are presented. Moreover, we demo…

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…

The closed string model in the background gravity field is considered as a bi-Hamiltonian system in assumption that string model is the integrable model for particular kind of the background fields. The dual nonlocal Poisson brackets(PB), depending of the background fields and of their derivatives, are obtained. The in…

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…

We develop a regularity theory for extremal knots of scale invariant knot energies defined by J. O'hara in 1991. This class contains as a special case the Möbius energy. For the Möbius energy, due to the celebrated work of Freedman, He, and Wang, we have a relatively good understanding. Their approch is crucially based…

Paper connects Bäcklund transformations to nonlocal pseudosymmetries.

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.

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.

We are concerned with unbounded sets of $\mathbb{R}^N$ whose boundary has constant nonlocal (or fractional) mean curvature, which we call CNMC sets. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We construct CNMC sets which are the countable union o…