Related papers: Busemann functions and barrier functions

We consider random walk in a space-time random potential, also known as directed random polymer measures, on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices. We construct covariant cocycles…

The comparison theory for the Riccati equation satisfied by the shape operator of parallel hypersurfaces is generalized to semi-Riemannian manifolds of arbitrary index, using one-sided bounds on the Riemann tensor which in the Riemannian…

We establish conditions for nonexistence of global solutions for a class of quasilinear parabolic problems with a potential on complete, non-compact Riemannian manifolds, including the Porous Medium Equation and the p-Laplacian with a…

Consider a BV function on a Riemannian manifold. What is its differential? And what about the Hessian of a convex function? These questions have clear answers in terms of (co)vector/matrix valued measures if the manifold is the Euclidean…

We consider a class of slow-fast processes on a connected complete Riemannian manifold $M$.The limiting dynamics as the scale separation goes to $\infty$ is governed by the averaging principle. Around this limit, we prove large deviation…

In this article, we study the large time behavior of solutions of first-order Hamilton-Jacobi Equations, set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish…

In this paper, we generalize a classical comparison result for solutions to Hamilton-Jacobi equations with Dirichlet boundary conditions, to solutions to Hamilton-Jacobi equations with non-zero boundary trace. As a consequence, we prove the…

We investigate the finiteness structure of a complete non-compact $n$-dimensional Riemannian manifold $M$ whose radial curvature at a base point of $M$ is bounded from below by that of a non-compact von Mangoldt surface of revolution with…

Assume $M$ is a closed, connected and smooth Riemannian manifold. We consider the evolutionary Hamilton-Jacobi equation \begin{equation*} \left\{ \begin{aligned} &\partial_t u(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\quad (x,t)\in…

In a Riemannian manifold with a smooth positive function that weights the associated Hausdorff measures we study stable sets, i.e., second order minima of the weighted perimeter under variations preserving the weighted volume. By assuming…

Let $\mathcal{M}$ be a Riemannian $n$-manifold with a metric such that the manifold is Ahlfors-regular. We also assume either non-negative Ricci curvature, or that the Ricci curvature is bounded from below together with a bound on the…

Motivated by Varadhan's theorem, we introduce Varadhan functions, variances, and means on compact Riemannian manifolds as smooth approximations to their Fr\'echet counterparts. Given independent and identically distributed samples, we prove…

These notes on Riemannian geometry use the bases bundle and frame bundle, as in Geometry of Manifolds, to express the geometric structures. It has more problems and omits the background material. It starts with the definition of Riemannian…

We study a family of gradient obstacle problems on a compact Riemannian manifold. We prove that the solutions of these free boundary problems are uniformly semiconcave and, as a consequence, we obtain some fine convergence results for the…

We consider Hamilton--Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results are the existence and well--posedness of a viscosity solution to the Cauchy problem. We define…

We study viscosity solutions to the classical one-phase problem and its thin counterpart. In low dimensions, we show that when the free boundary is the graph of a continuous function, the solution is the half-plane solution. This answers,…

This paper is devoted to a viscosity solution theory of the stochastic Hamilton-Jacobi-Bellman equation in the Wasserstein spaces for the mean-field type control problem which allows for random coefficients and may thus be non-Markovian.…

For a complete noncompact connected Riemannian manifold with bounded geometry $M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is a locally $\left(1-\frac{1}{n}\right)$-H\"older continuous function and so in particular it…

The interaction between thin structures and incompressible Newtonian fluids is ubiquitous both in nature and in industrial applications. In this paper we present an isogeometric formulation of such problems which exploits a boundary…

We prove several functional and geometric inequalities only assuming the linearity and a quantitative $\mathrm{L}^\infty$-to-Lipschitz smoothing of the heat semigroup in metric-measure spaces. Our results comprise a Buser inequality, a…