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

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 show that a class of Poincar\'e-Wirtinger inequalities on bounded convex sets can be obtained by means of the dynamical formulation of Optimal Transport. This is a consequence of a more general result valid for convex sets, possibly…

We prove invariant Harnack inequalities for certain classes of non-divergence form equations of Kolmogorov type. The operators we consider exhibit invariance properties with respect to a homogeneous Lie group structure. The coefficient…

We establish the boundedness of weak subsolutions for a class of degenerate Kolmogorov equations of hypoelliptic type, compatible with a homogeneous Lie group structure, within bounded product domains using the De Giorgi iteration. We…

We establish a new type of weak Harnack estimates with optimal parabolic tail for the weak supersolutions to a doubly nonlinear nonlocal $p$-Laplace equation, which is modeled on the nonlocal Trudinger equation. Our results are achieved by…

In 1957, De Giorgi [3] proved the H\"{o}lder continuity for elliptic equations in divergence form and Moser [7] gave a new proof in 1960. Next year, Moser [8] obtained the Harnack inequality. In this note, we point out that the Harnack…

In the Heisenberg group framework, we study rigidity properties for stable solutions of $(-\Delta_H)^s v = f(v)$ in $H$, $s \in (0,1)$. We obtain a Poincar\'e type inequality in connection with a degenerate elliptic equation in $\R^4_+$;…

This paper is devoted to studying the local behavior of non-negative weak solutions to the doubly non-linear parabolic equation \begin{equation*} \partial_t u^q - \text{div}\big(|D u|^{p-2}D u\big) = 0 \end{equation*} in a space-time…

We prove uniform parabolic H\"older estimates of De Giorgi-Nash-Moser type for sequences of minimizers of the functionals \[ \mathcal{E}_\varepsilon(W) = \int_0^\infty \frac{e^{- t/\varepsilon}}{\varepsilon} \bigg\{…

We show that weak solutions to parabolic equations in divergence form with zero Dirichlet boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order…

We prove second and fourth order improved Poincar\'e type inequalities on the hyperbolic space involving Hardy-type remainder terms. Since theirs l.h.s. only involve the radial part of the gradient or of the laplacian, they can be seen as…

The goal of this paper is to continue the study of the relation between the Poincar\'e inequality and the lower bounds of Minkowski content of separating sets, initiated in our previous work [Caputo, Cavallucci: Poincar\'e inequality and…

We deal with a wide class of generalized nonlocal $p$-Laplace equations, so-called nonlocal $G$-Laplace equations, in the Heisenberg framework. Under natural hypotheses on the $N$-function $G$, we provide a unified approach to investigate…

The main result of the paper is on the continuity of weak solutions of infinitely degenerate quasilinear second order equations. Namely, we show that every weak solution to a certain class of degenerate quasilinear equations is continuous.…

In this paper, we consider a weakly coupled system of nonlocal operators which contain both diffusion part with uniformly elliptic diffusion matrices and bounded drift vectors and the jump part with relatively general jump kernels. We use…

We study generalized fractional $p$-Laplacian equations to prove local boundedness and H\"older continuity of weak solutions to such nonlocal problems by finding a suitable fractional Sobolev-Poincar\'e inquality.

In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate when "the gradient is small". Typical examples are either equations involving the $m$-Laplace operator or Bellman-Isaacs equations…

We establish an isoperimetric type inequality for the level sets of functions in fractional Sobolev spaces. This answers a question posed by the first author in a previous paper. To obtain it, we work out a slight modification of some…

We study the regularity of weak solutions to nonlocal in time subdiffusion equations for a wide class of weakly singular kernels appearing in the generalised fractional derivative operator. We prove a weak Harnack inequality for nonnegative…

The weak Harnack inequality for $L^p$-viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that H\"older…