Related papers: Harnack Inequality on Homogeneous Spaces

We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong $A_\infty$ weight.…

We prove a boundary Harnack principle in Lipschitz domains with small constant for fully nonlinear and $p$-Laplace type equations with a right hand side, as well as for the Laplace equation on nontangentially accessible domains under extra…

We investigate discrete Poincar\'e inequalities on piecewise polynomial subspaces of the Sobolev spaces H(curl) and H(div) in three space dimensions. We characterize the dependence of the constants on the continuous-level constants, the…

We study relations and characterizations of various elliptic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces. We allow the scaling function be state-dependent and the state space possibly disconnected.…

We prove a scale-invariant boundary Harnack principle for inner uniform domains over a large family of Dirichlet spaces. A novel feature of our work is that our assumptions are robust to time changes of the corresponding diffusions. In…

The classical Poincar\'e inequality establishes that for any bounded regular domain $\Omega\subset \R^N$ there exists a constant $C=C(\Omega)>0$ such that $$ \int_{\Omega} |u|^2\, dx \leq C \int_{\Omega} |\nabla u|^2\, dx \ \ \forall u \in…

In this paper, we establish some Harnack type inequalities satisfied by positive solutions of nonlocal inhomogeneous equations arising in the description of various phenomena ranging from population dynamics to micro-magnetism. For regular…

We study regularity properties for solutions to the nakedly degenerate elliptic equation $a_{ij}\partial_{ij}u =0$, where the coefficients satisfy $I \ge a_{ij}(x) \ge \lambda(x) I$ and the only assumption is that $\lambda^{-1} \in L^p$. We…

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.

We present a unified strategy to derive Hardy-Poincar\'e inequalities on bounded and unbounded domains. The approach allows proving a general Hardy-Poincar\'e inequality from which the classical Poincar\'e and Hardy inequalities immediately…

In this paper we establish the Harnack inequality for globally positive local solutions to a general class of nonlocal in time subdiffusion equations in one space dimension, which includes time-fractional diffusion equations with time order…

We study local regularity properties for solutions of linear, non-uniformly elliptic equations. Assuming certain integrability conditions on the coefficient field, we prove local boundedness and Harnack inequality. The assumed integrability…

In this paper, we establish a scale invariant Harnack inequality for some inhomogeneous parabolic equations in a suitable intrinsic geometry dictated by the nonlinearity. The class of equations that we consider correspond to the parabolic…

We prove an improved version of Poincar\'e-Hardy inequality in suitable subspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As a consequence, we obtain a new Hardy type inequality with an improved constant (than the…

We investigate the dependence of optimal constants in Poincar\'e- Sobolev inequalities of planar domains on the region where the Dirichlet condition is imposed. More precisely, we look for the best Dirichlet regions, among closed and…

In this paper, the Harnack inequality result are established for a new class of the homogeneous nonlinear degenerate parabolic equations \begin{align*} div A(t,x,u,\nabla_x u)-\partial_t \vert u\vert^{p-2}u=0 \end{align*} on a bounded…

We study weak Harnack inequality and a priori H\"older regularity of harmonic functions for symmetric nonlocal Dirichlet forms on metric measure spaces with volume doubling condition. Our analysis relies on three main assumptions: the…

We prove a full Harnack inequality for local minimizers, as well as weak solutions to nonlocal problems with non-standard growth. The main auxiliary results are local boundedness and a weak Harnack inequality for functions in a…

Poincar\'{e}-Sobolev-type inequalities involving rearrangement-invariant norms on the entire $\mathbb{R}^n$ are provided. Namely, inequalities of the type $\|u-P\|_{Y(\mathbb{R}^n)}\leq C\|\nabla^m u\|_{X(\mathbb{R}^n)}$, where $X$ and $Y$…

Let $G$ be a real connected Lie group with polynomial volume growth, endowed with its Haar measure $dx$. Given a $C^2$ positive function $M$ on $G$, we give a sufficient condition for an $L^2$ Poincar\'e inequality with respect to the…