Related papers: Equivalent Harnack and Gradient Inequalities for P…

A new proof of the Wulff-Gage isoperimetric inequality for origin-symmetric convex bodies is provided. As its applications, we prove the uniqueness of log-Minkowski problem and a new proof of the log-Minkowski inequality of curvature…

In this paper we present an abstract convergence analysis of inexact descent methods in Riemannian context for functions satisfying Kurdyka-Lojasiewicz inequality. In particular, without any restrictive assumption about the sign of the…

We prove a Harnack inequality for positive solutions of a parabolic equation with slow anisotropic spatial diffusion. After identifying its natural scalings, we reduce the problem to a Fokker-Planck equation and construct a self-similar…

In this paper, we present a unified method for deriving differential Harnack inequalities for positive solutions of the semilinear parabolic equation \begin{equation*} \partial_t u=\Delta_V u+H(u) \end{equation*} on complete Riemannian…

We obtain a global extension of the classical weak Harnack inequality which extends and quantifies the Hopf-Oleinik boundary-point lemma, for uniformly elliptic equations in divergence form. Among the consequences is a boundary gradient…

We prove differential Harnack inequalities for flows of strictly convex hypersurfaces by powers $p$, $0<p<1$, of the mean curvature in Einstein manifolds with a positive lower bound on the sectional curvature. We assume that this lower…

In this paper we give both an historical and technical overview of the theory of Harnack inequalities for nonlinear parabolic equations in divergence form. We start reviewing the elliptic case with some of its variants and geometrical…

This article deals with kinetic Fokker-Planck equations with essentially bounded coefficients. A weak Harnack inequality for non-negative super-solutions is derived by considering their Log-transform and following S. N. Kruzhkov (1963).…

By the method of coupling and Girsanov transformation, Harnack inequalities [F.-Y. Wang, 1997] and strong Feller property are proved for the transition semigroup associated with the multivalued stochastic evolution equation on a Gelfand…

By constructing a suitable coupling by change of measures, the asymptotic log- Harnack inequality is established for a class of degenerate SPDEs with reflection. This inequality implies the asymptotic heat kernel estimate, the uniqueness of…

This paper extends the Bakry-\'{E}mery theorem connecting the Ricci curvature and log-Sobolev inequalities to the matrix-valued setting. Using tools from noncommuative geometry, it is shown that for a right invariant second order…

We give a short and self-contained proof of the Boundary Harnack inequality for a class of domains satisfying some geometric conditions given in terms of a state function that behaves as the distance function to the boundary, is subharmonic…

Let $M$ be a complete connected Riemannian manifold with boundary $\pp M$, $Q$ a bounded continuous function on $\pp M$, and $L= \DD+Z$ for a $C^1$-vector field $Z$ on $M$. By using the reflecting diffusion process generated by $L$ and its…

We derive logarithmic gradient estimate and universal boundedness estimate for semilinear elliptic equations on \RCD\, metric measure spaces, which contains the class of Riemannian manifolds with Ricci curvature bounded below. These…

On a Riemannian metric-measure space, we establish an Alexandrov-Bakelman-Pucci type measure estimate connecting Bakry-\'Emery Ricci curvature lower bound, modified Laplacian and the measure of certain special sets. We apply this estimate…

Being motivated by the problem of deducing $L^p$-bounds on the second fundamental form of an isometric immersion from $L^p$-bounds on its mean curvature vector field, we prove a (nonlinear) Calder\'on-Zygmund inequality for maps between…

We establish a lower bound on the total mass of the time slices of (n + 1)-dimensional asymptotically flat standard static spacetimes under the timelike convergence condition. The inequality can be viewed equivalently as a Minkowski-type…

We prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a Michael-Simon inequality for submanifolds in manifolds with nonnegative sectional curvature. Both inequalities depend on the asymptotic…

Inspired by the approach of Ivanisvili and Volberg towards functional inequalities for probability measures with strictly convex potentials, we investigate the relationship between curvature bounds in the sense of Bakry-Emery and local…

We prove a Heinz type inequality for harmonic diffeomorphisms of of the half-plane onto itself. We then apply this result to prove some sharp bound of the Gaussian curvature of a minimal surface, provided that it lies above the whole…