Related papers: H\"older regularity for weak solutions to nonlocal…
In this article, we study the following nonlinear doubly nonlocal problem involving the fractional Laplacian in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{\begin{aligned} (-\Delta)^s u & =…
We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical ($\alpha <1/2$) dissipation $(-\Delta)^\alpha$. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study…
Minima of the log-multiphase variational integral $$ w \mapsto \int_{\Omega} \left[|Dw|\log(1+|Dw|) + a(x)|Dw|^q + b(x)|Dw|^s\right] \, {\rm d}x\,, $$ have locally H\"older continuous gradient under sharp quantitative bounds linking the…
We study the local H\"older continuity of nonnegative solutions to doubly nonlinear equations by introducing a new technique that allows us to treat the cases where the equation is both singular and degenerate, up to specific Barenblatt…
We prove a higher regularity result for weak solutions to nonlinear nonlocal equations along the integrability scale of Bessel potential spaces $H^{s,p}$ under a mild continuity assumption on the kernel. By embedding, this also yields…
In this paper, we develop some properties of the $a_{x,y}(\cdot)$-Neumann derivative for the nonlocal $s(\cdot,\cdot)$-order operator in fractional Musielak-Sobolev spaces with variable $s(\cdot,\cdot)-$order. Therefore we prove the basic…
We study the existence, multiplicity and regularity results of non-negative solutions of following doubly nonlocal problem: $$ (P_\la) \left\{ \begin{array}{lr}\ds \quad (-\Delta)^{s_1}u+\ba (-\Delta)^{s_2}_{p}u = \la a(x)|u|^{q-2}u+…
Motivated by problems arising in geometric flows, we prove several regularity results for systems of local and nonlocal equations, adapting to the parabolic case a neat argument due to Caffarelli. The geometric motivation of this work comes…
We prove that $L^2$ weak solutions to hypoelliptic equations with bounded measurable coefficients are H\"older continuous. The proof relies on classical techniques developed by De Giorgi and Moser together with the averaging lemma and…
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…
Let $2\le n\le 5$. We establish an apriori interior H\"older regularity of $C^2$-stable solutions to the semilinear equation $-\Delta u=f(u)$ in any domain of $R^n$ for any nonlinearity $f\in C^{0,1}(R) $.If $f $ is nondecreasing and convex…
The aim of this paper is to establish regularity for weak solutions to the nondiagonal quasilinear degenerate elliptic systems related to H\"{o}rmander's vector fields, where the coefficients are bounded with vanishing mean oscillation. We…
We consider equations involving a combination of local and nonlocal degenerate $p$-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and H\"older continuity with an explicit…
This paper is concerned with H\"older regularity of viscosity solutions of second-order, fully non-linear elliptic integro-differential equations. Our results rely on two key ingredients: first we assume that, at each point of the domain,…
We consider mixed local and nonlocal quasilinear parabolic equations of $p$-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions,…
We present some new regularity criteria for ``suitable weak solutions'' of the Navier-Stokes equations near the boundary in dimension three. We prove that suitable weak solutions are H\"older continuous up to the boundary provided that the…
In this paper, we study the regularity of weak solutions to the following strongly degenerate parabolic equation \begin{equation*} u_t-\div\left(\left(\left|Du\right|-1\right)_+^{p-1}\frac{Du}{\left|Du\right|}\right)=f\qquad\mbox{ in…
In this paper we study some boundary value problems for a fractional analogue of second order elliptic equation with an involution perturbation in a rectangular domain. Theorems on existence and uniqueness of a solution of the considered…
We investigate local regularity properties of weak solutions to a broad class of nonlinear nonlocal kinetic Kolmogorov-Fokker-Planck equations. In particular, we focus on proving an interpolative apriori boundedness estimate for weak…
In this paper, we are concerned with the partial regularity of the suitable weak solutions to the fractional MHD equations in $\mathbb{R}^{n}$ for $n=2,\,3$. In comparison with the work of the 3D fractional Navier-Stokes equations obtained…