Related papers: Poincar\'e inequality and quantitative De Giorgi m…

We study H\"older continuity for solutions of the $p$-Laplace equation. This is established through a method involving an ordinary differential inequality, in contrast to the classical proof of the De Giorgi-Nash-Moser Theorem which uses…

We establish weak Harnack inequalities for positive, weak supersolutions to certain doubly degenerate parabolic equations. The prototype of this kind of equations is $$\partial_tu-\operatorname{div}|u|^{m-1}|Du|^{p-2}Du=0,\quad p>2,\quad…

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincar\'{e} inequalities, general Beckner inequalities...). We also discuss the…

We give a proof of H\"older continuity for bounded local weak solutions to the equation $u_t= \sum_{i=1}^N (|u_{x_i}|^{p_i-2} u_{x_i})_{x_i}$, in $\Omega \times [0,T]$, with $\Omega \subset \subset \mathbb{R}^N$, under the condition $…

We prove the local H\"older regularity of weak solutions to the mixed local nonlocal parabolic equation of the form \begin{equation*} u_t-\Delta u+\text{P.V.}\int_{\mathbb{R}^{n}} {\frac{u(x,t)-u(y,t)}{{\left|x-y\right|}^{n+2s}}}dy=0,…

We give a generalization of a theorem of B\^ocher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a…

We prove a Harnack inequality for the solutions of a difference equation with non-elliptic balanced i.i.d. coefficients. Along the way we prove a (weak) quantitative homogenisation result, which we believe is of some interest too.

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).…

This paper is dedicated to the application of the DeGiorgi-Nash-Moser regularity theory to the kinetic Fokker-Planck equation. This equation is hypoelliptic. It is parabolic only in the velocity variable, while the Liouville transport…

In this article, we establish radial symmetry for positive weak solutions of a class of mixed local-nonlocal equations with possibly singular nonlinearity via the moving plane method. Furthermore, we provide a quantitative version of…

In this short note, we provide a quantitative global Poincar\'e inequality for one forms on a closed Riemannian four manifold, in terms of an upper bound on the diameter, a positive lower bound on the volume, and a two-sided bound on Ricci…

We prove local H\"older continuity for non negative, locally bounded, local weak solutions to the class of doubly nonlinear parabolic equations $\partial_t (u_q) - \text{div} (|Du|^{p-2} Du) = 0$ for $p > 2$, $ 0 < q < p-1$. The proof…

We establish the interior and boundary H\"older continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{p-2}u\big)-\Delta_p u=0,\quad p>1. \] The proof…

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…

The aim of this article is to develop the regularity theory for parabolic equations driven by nonlocal operators associated with nonsymmetric forms. H\"older regularity and weak Harnack inequalities are proved using extensions of recently…

This paper is devoted to a qualitative analysis of the Poincar\'e--Sobolev level associated with the fractional GJMS operators \(\mathcal{P}_s\) \(\bigl(s\in(0,\tfrac n2)\setminus\mathbb N\bigr)\) on the hyperbolic space \(\mathbb H^n\). In…

We study viscosity solutions to degenerate and singular elliptic equations of $p$-Laplacian type on Riemannian manifolds. The Krylov-Safonov type Harnack inequality for the $p$-Laplacian operators with $1<p<\infty$ is established on the…

We study parabolic equations governed by integro-differential operators with nonlocal components in some directions and local components in the remaining directions. The setting contains the purely nonlocal, as well as the purely local…

In this paper, we obtain the reverse Bakry-\'Emery type estimates for a class of hypoelliptic diffusion operator by coupling method. The (right and reverse) Poincar\'e inequalities and the (right and reverse) logarithmic Sobolev…

We extend the celebrate De Giorgi-Nash-Moser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional $p$-Laplacian operator on the Heisenberg-Weyl…