The paper proves a nonlocal version of the Alexandrov Theorem for smooth boundaries.
problem Proving the nonlocal version of the Alexandrov Theorem for sets with smooth boundaries.
method Formulated a necessary and sufficient condition for the theorem to hold, used a specific formula for the tangential derivative of the nonlocal mean curvature, and applied the method of moving planes.
result The only set with smooth boundary and constant nonlocal mean curvature is an Euclidean ball.
Study of nonlocal mean curvature and symmetry of surfaces.
problem Symmetry of surfaces with ordered nonlocal mean curvature.
method Generalization of Alexandrov's moving plane method.
result Similar result to classical setting proved in nonlocal setting.
Mathematicians study surfaces with constant nonlocal mean curvature.
problem Overdetermined boundary value problems.
method Properties of Nonlocal Mean Curvature and related differential operators.
result Surfaces with constant NMC have a close connection to overdetermined problems.
We are concerned with hypersurfaces of RN 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…
Smoothness of graphs evolving by fractional mean curvature is proven.
problem Evolution of graphs by fractional mean curvature.
method Analytic semigroup approach to nonlocal quasilinear evolution equation.
result Short time existence, uniqueness, and optimal Hölder regularity of classical solutions.
We study hypersurfaces of RN 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 RN, N≥2, all of th…
First variation of fractional k-dimensional measure for submanifolds
problem Computing the first variation of a fractional k-dimensional measure for submanifolds method First variation computation
result Definition of a nonlocal mean-curvature vector for embedded submanifolds
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 C2-distance fro…
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.
Estimates for VPMCF show ancient MCF solutions and finite-time behavior.
problem Volume Preserving Mean Curvature Flow (VPMCF) behavior and singularities.
method Nonlocal estimates and blowup analysis.
result Ancient solutions to MCF and finite-time behavior of VPMCF.
Introduces fractional length and nonlocal curvature for smooth curves.
problem Defining curvature for curves of fractional length.
method Introduces fractional length and derives nonlocal curvature using fractional perimeter analogy.
result Fractional length converges to traditional length with a multiplicative constant.
The study proves that sets with constant nonlocal curvature are composed of equal balls under certain conditions.
problem Characterizing sets with constant nonlocal curvature.
method Analyzing measurable sets in R^d with constant nonlocal h-mean curvature under a suitable integrability assumption.
result Finite unions of equal balls are the only sets with constant nonlocal curvature under the given conditions.
Study on sets with constant nonlocal mean curvature in bounded domains.
problem Localization of sets with constant nonlocal mean curvature in bounded domains.
method Proving sets are close to critical points of a non-local potential, existence of minimizers under fixed volume constraints.
result Existence and properties of minimizers in half-spaces, including smoothness and symmetry.
Gradient estimate for nonlocal minimal graphs in higher dimensions.
problem Establishing a gradient bound for nonlocal minimal graphs in higher dimensions.
method Using a truncated fractional Jacobi operator and a weak Harnack inequality, the gradient bound is derived.
result Gradient of nonlocal minimal graphs is bounded by a power of their oscillation.
Approximates nonlocal curvature of curves using splines.
problem Approximating nonlocal curvature of planar curves.
method Extending nonlocal curvature definition, using incomplete beta function, and linear interpolating spline approximation.
result Nonlocal curvature of a planar curve can be approximated by a spline.
Proposes a new nonlocal curvature tensor concept.
problem Various nonlocal curvature concepts in literature.
method Generalizes classical curvature tensor representation and uses fractional differential operator analogies.
result Introduces a new nonlocal curvature tensor.
We study the phase field method for the volume preserving mean curvature flow. Given an initial C1 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…
New connected sum method for zero scalar curvature with constant mean curvature boundary.
problem Prescribing zero scalar curvature with constant mean curvature boundary on connected sums of manifolds.
method Boundary connected sum construction, exploiting nonlocal aspects and recent tools.
result Construction of a connected sum with zero scalar curvature and constant mean curvature boundary.
We are concerned with unbounded sets of RN 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…
The study proves inequalities for complex operators on curved spaces.
problem Establishing inequalities for nonlocal operators on curved spaces.
method Defining and analyzing nonlocal Pucci operators on manifolds with nonnegative sectional curvatures, proving Harnack inequalities and Holder estimates.
result Harnack inequalities and Holder estimates for nonlocal operators on manifolds with nonnegative sectional curvatures.
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 Rn evolving by fractional mean curvature flow, which developes a singularity before it can shrink to a point. When n>3 this resul…
Proves uniqueness of solutions for a nonlocal Liouville equation with finite Q-curvature.
problem Proving uniqueness of solutions for a specific nonlocal Liouville equation.
method Connection to Calogero--Moser derivative NLS and ground state solitons.
result Uniqueness of solutions in the Gaussian case and general positive, symmetric-decreasing K. Study finds conditions for global minimizers on curved manifolds with fast diffusion and nonlocal interactions.
problem Existence of global minimizers for a free energy functional on negatively curved manifolds.
method Investigation of Carlson-Levin type inequalities for Cartan-Hadamard manifolds.
result Establishes necessary and sufficient conditions for the existence of global energy minimizers.
Flow of planar curves with curvature and capacity potential.
problem Geometric flow of planar curves with curvature and capacity potential.
method Curvature flow with a nonlocal term (normal derivative of capacity potential).
result Long term existence and large time asymptotics established under convexity condition.
Study uses SAR data to map defoliation and regrowth in tundra-forest areas.
problem Mapping defoliation and regrowth in tundra-forest areas using SAR data.
method Novel guided nonlocal means speckle filtering of polarimetric covariance matrix.
result Over 99.7% classification accuracy in defoliation and regrowth mapping.
New algorithm solves complex mean-field Schrödinger bridge problem.
problem Designing a controller for diffusion processes with nonlocal interaction.
method Generalized Hopf-Cole transform and Sinkhorn-type algorithm.
result Convergence guarantees for the proposed algorithm under mild assumptions.
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…
Study on dynamic curves with elastic energy and spontaneous curvature.
problem Modeling and analyzing dynamic planar curves with elastic energy.
method Gradient flow of inclination angle, nonlocal quasilinear system, local well-posedness, global existence, convergence.
result Local well-posedness, global existence, convergence of the flow for weak regularity initial data.
Novel weak solutions for volume-preserving mean curvature flow established.
problem Existence and uniqueness of solutions to volume-preserving mean curvature flow.
method Introducing varifold solutions coupled with phase volumes and new calibrations.
result Uniqueness of classical solutions among varifold solutions.
Study dynamic asset allocation in incomplete markets using game theory and nonlocal BSDEs.
problem Dynamic mean-variance asset allocation in general incomplete markets with non-exponential discounting.
method Game-theoretic approach, decomposition into myopic and hedging strategies, nonlocal BSDEs, fixed-point theorem.
result Well-posedness of solutions to BSDEs, existence of equilibrium control policy.
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 capillarity minimizers with nonlocal repulsion and gravity, proving existence and nonexistence.
problem Minimizing an energy functional with capillarity, nonlocal repulsion, and gravity.
method Quantitative isoperimetric inequalities applied to capillarity problem in a half-space.
result Existence and nonexistence of minimizers for various nonlocal kernels and masses.
Study on consensus formation in manifolds with curvature constraints.
problem Long-time behavior of solutions to nonlocal PDEs on Riemannian manifolds.
method Analytical and numerical methods applied to self-collective models.
result Sufficient conditions for consensus formation and convergence rates quantified.
New nonlocal neural network learns stable dynamics for deeper nonlocal structures.
problem Capturing long-range dependencies in feature space.
method Spectrum analysis on weight matrices, new nonlocal block formulation.
result Stable dynamics in deeper nonlocal structures.
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…
The paper proves a Fenchel theorem for Gauss maps and shows circles and disks minimize certain energies.
problem Finding minimizers of nonlocal curvature energies.
method Combining Fenchel-type theorems with geometric analysis techniques.
result Circles and disks minimize specific energy functionals.
In this paper, we establish that: Suppose a closed Riemannian manifold (Mn,g0) of dimension ≥8 is not locally conformally flat, then the Paneitz-Sobolev constant of Mn has the property that q(g0)<q(Sn). 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…
Investigates convexity of minimizers under mass constraint using nonlocal perimeter and potential.
problem Convexity of minimizers under mass constraint.
method Nonlocal free energy with nonlocal perimeter and convex potential.
result Quantitative stability theorem for nonlocal free energy assuming symmetry on the potential.
Flow deforms locally convex curves to curves of constant k-order width.
problem Evolve locally convex curves to curves of constant k-order width.
method Introduced a nonlocal curvature flow to evolve locally convex curves in the plane.
result The flow converges to a smooth, locally convex curve of constant k-order width as time goes to infinity.
The Darboux-Egoroff system of PDEs with any number n≥3 of independent variables plays an essential role in the problems of describing n-dimensional flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a recursion operator and its inverse for symmetries of the Darboux-Egoroff system and des…
Survey discusses recent advances on Brakke flows and their properties.
problem Existence and regularity of Brakke flows.
method Analyzes recent progress including new theorems and proofs.
result Proof that branching singularities trigger dynamical instability.
Paper connects Bäcklund transformations to nonlocal pseudosymmetries.
problem Finding Bäcklund transformations for differential equations.
method Factorization with nonlocal pseudosymmetries to determine Bäcklund transformations as C-morphisms. result Bäcklund transformations are determined by nonlocal pseudosymmetries' invariants.
The paper proves removable singularity for nonlocal minimal graphs.
problem Proving removable singularities for nonlocal minimal graphs.
method Analyzing (s,1)-capacity zero compact sets to ensure graphs are minimal in the entire domain. result Nonlocal minimal graphs are removable in the entire domain if they are minimal in a set of (s,1)-capacity zero. The paper studies nonlocal isoperimetric problems in hyperbolic space and finds unique minimizers for small volumes.
problem Nonlocal isoperimetric problem in hyperbolic space.
method Investigates minimization of a functional with perimeter and a nonlocal term derived from the negative power of distance.
result Geodesic balls are unique minimizers for small volumes in hyperbolic space.
Study on evolving interfaces with complex curvature and density effects.
problem Understanding the dynamics of evolving heterogeneous elastic interfaces.
method Modeling an evolving curve with a density function, analyzing the associated gradient flow evolution.
result Analysis of the preservation and asymptotic behavior of geometric properties in the evolving system.
Paper develops new methods linking quantum mechanics and finance.
problem Modeling financial markets using quantum mechanics.
method Inverts quantum stochastic processes to nonlocal diffusions, uses path integral formalism and PT symmetry.
result Non-Gaussian kernel function for Accardi-Boukas quantum Black-Scholes.
In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in S1,} \end{equation} where (−Δ)21 stands for the fractional Laplacian and κ is a bounded function. We interpret the above equation as the prescri…