arXiv research

Geometric analysis proves weak KAM solutions constant under specific conditions.

In this paper, we strengthen the splitting theorem proved in [14, 15] and provide a different approach using ideas from the weak KAM theory.

We review and extend here some recent results on the existence of minimal surfaces and isoperimetric sets in non homogeneous and anisotropic periodic media. We also describe the qualitative properties of the homogenized surface tension, also known as stable norm (or minimal action) in Weak KAM theory. In particular we …

Global singularities propagate in magnetic mechanical systems on Riemannian manifolds.

The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws a lot of attention. …

For Finsler metrics (no reversibility assumed) on closed orientable surfaces of genus greater than one, we study the dynamics of minimal rays and minimal geodesics in the universal cover. We prove in particular, that for almost all asymptotic directions the minimal rays with these directions laminate the universal cove…

Let $M$ be a closed oriented surface of negative Gaussian curvature and let $Ω$ be a non-exact 2-form. Let $λ$ be a small positive real number. We show that the longitudinal KAM-cocycle of the magnetic flow given by $\la Ω$ is a coboundary if and only if the Gaussian curvature is constant and $Ω$ is a constant multiple…

For an integrable Hamiltonian with $d\ (d\geq 2)$ degrees of freedom, we show the conditions on perturbations, for which invariant tori can be destructed.

Normal forms prove dynamical results for magnetic fields on surfaces.

Let $M$ be a closed oriented surface and let $Ω$ be a non-exact 2-form. Suppose that the magnetic flow $φ$ of the pair $(g,Ω)$ is Anosov. We show that the longitudinal KAM-cocycle of $φ$ is a coboundary if and only the Gaussian curvature is constant and $Ω$ is a constant multiple of the area form thus extending the res…

Study shows invariant curves in tubular origami dynamics, revealing geometric barriers to folding transitions.

We study the structure of the stable norm of Finsler metrics on the 2-torus with a focus to points of irrational slope. By our results, the stable norm detects KAM-tori and hyperbolicity in the geodesic flow. Moreover, we study the stable norm in some natural examples.

Proves existence and uniqueness of weak solutions for specific equations.

Paper proves uniqueness of weak solutions for Plateau flow.

New metrics reveal how Majoranas crystallize in a particle model.

Study finds weak solutions for complex map flows with optimal lifespan.

Gradient estimates for solutions to a p-Laplacian equation on Riemannian manifolds.

Wire billiard is defined by a smooth embedded closed curve of non-vanishing curvature $k$ in $\mathbb{R}^n$ (a wire). For a class of curves, that we call nice wires, the wire billiard map is area preserving twist map of the cylinder. In this paper we are investigating whether the basic features of conventional planar b…

Novel weak solutions for volume-preserving mean curvature flow established.

We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.

We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface $\partialΩ_0$, we show that there exists a weak solution to the null mean curvatu…

New varifold solutions for mean curvature flow converge and are unique.

Alternative proof of weak solutions to mean curvature flow using minimizing movements.

Study constructs solutions for evolving hypersurfaces using inverse spacetime mean curvature.

The system of weak normality equations constitutes a part in the complete system of normality equations. Solutions of each of these two systems of equations are associated with some definite classes of Newtonian dynamical systems in Riemannian manifolds. In this paper for the case of simplest flat Riemannian manifold $…

The paper examines partial regularity of Lipschitz solutions to minimal surface system.

New boundary condition for weak inverse mean curvature flow in bounded domains.

Faster weak supervision framework using triplet methods.

We study weak solutions to degenerate quasilinear elliptic equations, involving first order terms, in unbounded tubular domains. In particular we show that, under suitable hypotheses, the weak comparison principle holds if the domain is narrow enough.

Study on fourth order Lamm-Riviere system for biharmonic mappings in 4D.

We prove the existence of weak solutions of complex $m-$Hessian equations on compact Hermitian manifolds for the nonnegative right hand side belonging to $L^p, p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the equation has been recently solved by Szekelyhidi and Zhang. We also give a stabilit…

Proves existence of proper solutions for inverse mean curvature flow.

In this paper, we discuss a Donaldson's version of the modified $K$-energy associated to the Calabi's extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified $K$-energy.

Paper explores weak solutions' regularity in critical dimensions without conservation law.

Study shows how weak inverse anisotropic mean curvature flow behaves at infinity.

Develops weak PINNs for efficient manifold solutions of hyperbolic equations.

Paper proves rigidity of weak solutions for anisotropic N-Laplacian equations with Neumann or Robin boundary conditions.

Proves higher regularity for anisotropic inverse mean curvature flow.

We investigate a parabolic-elliptic system which is related to a harmonic map from a compact Riemann surface with a smooth boundary into a Lorentzian manifold with a warped product metric. We prove that there exists a unique global weak solution for this system which is regular except for at most finitely many singular…

Paper reconciles different Ricci flow approaches and proves weak solutions.

We prove the existence of a unique global weak solution to the full bosonic string heat flow from closed Riemannian surfaces to an arbitrary target under smallness conditions on the two-form and the scalar potential. The solution is smooth with the exception of finitely many singular points. Finally, we discuss the con…

We study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove the global existence of the weak solutions for the mean curvature flow with the terms, by using the modified Allen-Cahn equation that holds useful properties such as the monotonicity formula.

In this note, we prove the existence of weak solutions of the Chern-Ricci flow through blow downs of exceptional curves, as well as backwards smooth convergence away from the exceptional curves on compact complex surfaces. The smoothing property for the Chern-Ricci flow is also obtained on compact Hermitian manifolds o…

New approach finds solutions to games with unbounded controls.

Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.

Study optimal transport on globally hyperbolic spacetimes, focusing on weak Kantorovich potentials' regularity.

Flat isometric immersions in 3D are developable if they are $C^{1,2/3}$ regular.

This paper concerns the inverse mean curvature flow of convex hypersurfaces which are Lipschitz in general. After defining a weak solution, we study the evolution of the singularity by looking at the blow-up tangent cone around each singular point. We prove the cone also evolves by the inverse mean curvature flow and e…