Related papers: De Giorgi-Nash-Moser theory for kinetic equations …
We report on recent results and a new line of research at the crossroad of two major theories in the analysis of partial differential equations. The celebrated De Giorgi-Nash-Moser theory shows H{\"o}lder estimates and the Harnack…
We derive quantitatively the Harnack inequalities for kinetic integro-differential equations. This implies H\"older continuity. Our method is based on trajectories and exploits a term arising due to the non-locality in the energy estimate.…
We consider a combination of local and nonlocal $p$-Laplace equations and discuss several regularity properties of weak solutions. More precisely, we establish local boundedness of weak subsolutions, local H\"older continuity of weak…
We derive the Strong Harnack inequality for a class of hypoelliptic integro-differential equations in divergence form. The proof is based on a priori estimates, and as such extends the first non-stochastic approach of the non-local…
In this paper, we investigate the regularity for mixed local and nonlocal degenerate elliptic equations in the Heisenberg group. Inspired by the De Giorgi-Nash-Moser theory, the local boundedness of weak subsolutions and the H\"{o}lder…
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…
In this paper, we establish the De Giorgi-Nash-Moser theory for a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, with the fractional $p$-Laplacian operator in Grushin-type spaces…
We consider hypoelliptic equations of kinetic Fokker-Planck type, also known as Kolmogorov or ultraparabolic equations, with rough coefficients in the drift-diffusion operator. We give novel short quantitative proofs of the De Giorgi…
In this paper, by applying the De Giorgi-Nash-Moser theory we prove nonlocal Harnack inequalities for (locally nonnegative in $\Omega$) weak solutions to nolocal double phase equations \begin{equation*}\begin{cases}\cL u =0 & \text{ in…
This article addresses the local boundedness and H\"older continuity of weak solutions to kinetic Fokker-Planck equations with general transport operators and rough coefficients. These results are due to the mixing effect of diffusion and…
We present a formalization in Lean of the core interior De Giorgi--Nash--Moser theory for uniformly elliptic divergence-form equations with bounded measurable coefficients. The formalized results include local boundedness of weak…
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…
A set of pointwise estimates are established for local solutions to nonlocal diffusion equations with a drift term. In particular, our Harnack estimates are the first ones for such equations, and our H\"older regularity refines certain…
We study the local H\"{o}lder regularity of weak solutions to the fully fractional parabolic equations involving spatial fractional diffusion and fractional time derivatives of the Marchaud type. It is worth noting that we do not impose…
We consider nonlinear nonlocal diffusive evolution equations, governed by fractional Laplace-type operators, fractional time derivative and involving porous medium type nonlinearities. Existence and uniqueness of weak solutions are…
We consider the following class of mixed local-nonlocal equations: \begin{align}\label{abs}\tag{$\mathcal{P}$} -\Delta_p u + (-\Delta)_p^s u = V |u|^{p-2}u \text{ in } \Omega, \end{align} where $s \in (0,1), p \in (1, \infty)$, and the…
In this article, we give a trajectorial proof of a kinetic Poincar\'e inequality which plays an important role in the De Giorgi-Nash-Moser theory for kinetic equations. The present work improves a result due to J. Guerand and C. Mouhot [10]…
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…
Local boundedness and Harnack inequalities are studied for solutions to parabolic and elliptic integro-differential equations whose governing nonlocal operators are associated with nonsymmetric forms. We present two independent proofs, one…
We study the local behavior of the elements of a specific energy class of functions, called the nonlocal parabolic ($p$-homogenous) De Giorgi class. First we carry on an analysis of their local boundedness under optimal tail conditions, and…