Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.0017

A Morse type uniqueness theorem for non-parametric minimizing hypersurfaces

Hannes Junginger-Gestrich
View original source ↗Read source document ↗Export metadata ↓

Abstract

A classical result about minimal geodesics on R^2 with Z^2 periodic metric that goes back to H.M. Morse asserts that a minimal geodesic that is asymptotic to a periodic minimal geodesic cannot intersect any periodic minimal geodesic of the same period. This paper treats a similar theorem for nonparametric minimizing hypersurfaces without selfintersections -- as were studied by J. Moser, V. Bangert, P.H. Rabinowitz, E. Stredulinsky and others.

Explore related subjects

math.APmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hannes Junginger-Gestrich. 2007-07-02. A Morse type uniqueness theorem for non-parametric minimizing hypersurfaces. https://arxiv.org/abs/0707.0017

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Global higher integrability and Hardy inequalities for double-phase functionals under a capacity density condition

We prove global higher integrability for functionals of double-phase type under a uniform local capacity density condition on the complement of the considered domain $Ω\subset \mathbb{R}^n$. In this context, we investigate a new natural notion of variational capacity associated to the double-phase integrand. Under the related fatness condition for the complement of $Ω$, we establish an integral Hardy inequality. Further, we show that fatness of $\mathbb{R}^n \setminus Ω$ is equivalent to a boundary Poincaré inequality, a pointwise Hardy inequality and to the local uniform $p$-fatness of $\mathbb{R}^n \setminus Ω$. We provide a counterexample that shows that the expected Maz'ya type inequality - a key intermediate step toward global higher integrability - does not hold with the notion of capacity involving the double-phase functional itself.

math.AP↗
arXiv2026

Modulus of continuity for solutions of non-local heat equations

We extend the two-point modulus-of-continuity method for parabolic equations---as used, for instance, in the proof of the fundamental gap conjecture---to non-local heat equations on $\R^n$. An initial modulus of continuity is shown to be preserved provided that its odd extension is a supersolution of an associated one-dimensional non-local equation. The result applies to both integrable and singular kernels, including the fractional Laplacian, and the method further applies to a class of nonlinear non-local heat equations on $\R^n$. We also examine the corresponding regional problem on bounded convex sets. The natural distance-only modulus comparison fails: a smooth counterexample is constructed in one dimension, and the same obstruction is demonstrated to persist in higher dimensions. For the regional fractional Laplacian, we define convex domains of fixed diameter for which the first non-trivial eigenvalue tends to zero. There is consequently no positive Payne--Weinberger-type lower bound for this eigenvalue depending only on the diameter of the domain.

math.AP↗
arXiv2026

Long time behavior of discrete velocity kinetic equations

We study long time behavior of some nonlinear discrete velocity kinetic equations in the one and three dimensions with periodic boundary conditions. We prove the exponential time decay of solutions towards the global equilibrium in the $L^2$ space. Our result holds for a wide class of interaction rates including the Goldstein-Taylor and Carleman equations, and the estimates on the rate of convergence are explicit and constructive. The technique is based on the construction of suitable Lyapunov functionals by modifying Boltzmann's entropy.

math.AP↗