Related papers: On geodesics in low regularity

We prove some regularity results for a priori bounded local minimizers of non-autonomous integral functionals of the form $$\mathcal{F}(v,\Omega)=\int_\Omega F(x,Dv)dx,$$ under the constraint $v \ge \psi$ a.e. in $\Omega$, where $\psi$ is a…

Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min-max theory which we develop in this article for the free boundary variational…

This paper deals with the Lipschitz regularity of minimizers for a class of variational obstacle problems with possible occurance of the Lavrentiev phenomenon. In order to overcome this problem, the availment of the notions of relaxed…

We use the method of layer potentials to study the $R_2$ Regularity problem and the $D_2$ Dirichlet problem for second order elliptic equations of the form $\mathcal{L}u=0$, with lower order coefficients, in bounded Lipschitz domains. For…

We study geometric and topological properties of locally compact, geodesically complete spaces with an upper curvature bound. We control the size of singular subsets, discuss homotopical and measure-theoretic stratifications and regularity…

Minimal surfaces in the Riemannian product of surfaces of constant curvature have been considered recently, particularly as these products arise as spaces of oriented geodesics of 3-dimensional space-forms. This papers considers more…

We consider epimorphisms and several variant notions -- split, effective, regular, strong, and extremal -- and determine which of these coincide in the metric coarse and coarsely Lipschitz categories. In particular, we characterise extremal…

We prove local Lipschitz regularity for local minimiser of \[ W^{1,1}(\Omega)\ni v\mapsto \int_\Omega F(Dv)\, dx \] where $\Omega\subseteq {\mathbb R}^N$, $N\ge 2$ and $F:{\mathbb R}^N\to {\mathbb R}$ is a quasiuniformly convex integrand in…

This paper is devoted to the regularity analysis of a geodesic equation in the space of Sasakian metrics. Firstly, we reduce the geodesic equation in the space of Sasakian metrics to a Dirichlet problem of degenerate complex Monge-Amp\'ere…

We investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as well as with singular Alexandrov surfaces…

We discuss the problem of deciding when a metrisable topological group $G$ has a canonically defined local Lipschitz geometry. This naturally leads to the concept of minimal metrics on $G$, that we characterise intrinsically in terms of a…

The present article investigates the existence, multiplicity and regularity of weak solutions of problems involving a combination of critical Hartree type nonlinearity along with singular and discontinuous nonlinearity. By applying…

The aim of this paper is to extend the definition of geodesics to conical manifolds, defined as submanifolds of $\R^n$ with a finite number of singularities. We look for an approach suitable both for the local geodesic problem and for the…

We study inverse boundary problems for semilinear Schr\"odinger equations on smooth compact Riemannian manifolds of dimensions $\ge 2$ with smooth boundary, at a large fixed frequency. We show that certain classes of cubic nonlinearities…

Regularity theory for diffusive operators is among the finest treasures of the modern mathematical sciences. It appears in several different fields, such as, differential geometry, topology, numerical analysis, dynamical systems,…

The goal of this paper is to study periodic geodesics for sub-Riemannian metrics on a contact 3D-manifold.We develop two rather independent subjects:1) The existence of closed geodesics spiraling around periodic Reeb orbits for a generic…

We investigate the rudiments of Riemannian geometry on orbit spaces $M/G$ for isometric proper actions of Lie groups on Riemannian manifolds. Minimal geodesic arcs are length minimising curves in the metric space $M/G$ and they can hit…

We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces and prove the nonexistence of…

The geometric approach to optimal transport and information theory has triggered the interpretation of probability densities as an infinite-dimensional Riemannian manifold. The most studied Riemannian structures are Otto's metric, yielding…

In this paper, we study the regularity of weak solutions and subsolutions of second-order elliptic equations having a gradient term with superquadratic growth. We show that, under appropriate integrability conditions on the data, all weak…