Related papers: $W^{\sigma,\epsilon}$-estimates for nonlocal ellip…

We discuss unique existence and microlocal regularity properties of Sobolev space solutions to the fractional Eringen wave equation, initially given in the form of a system of equations in which the classical non-local Eringen constitutive…

We study integrability of the derivative of solutions to a singular one-dimensional parabolic equation with initial data in $W^{1,1}$. In order to avoid additional difficulties we consider only the periodic boundary conditions. The problem…

In this work, we establish global gradient estimates to solutions of quasilinear elliptic models in non-divergence form with general degeneracy law and a Hamiltonian term, given by $$ -\Psi(x, |\nabla…

In this work, we shall investigate existence and multiplicity of solutions for a nonlocal elliptic systems driven by the fractional Laplacian. Specifically, we establish the existence of two positive solutions for following class of…

We establish derivative estimates of solution of elliptic system in narrow regions.

Recently, great attention has been focused on the study of fractional and non-local operators of elliptic type, both for pure mathematical research and in view of concrete real-world applications. Our problem is related to the fractional…

In this paper we consider the existence and multiplicity of weak solutions for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-\Delta)^{\frac{1}{2}}u + u &= Q(x)f(u)\;\;\mbox{in}\;\;\R…

This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{equation*}\label{00} \left\{ \begin{array}{l} (-\Delta)^{s}u + u = |u|^{p-2}u\;\;\mbox{in $\Omega$},\\ u \geq 0 \quad \mbox{in}…

We consider local weak solutions to the fractional $p$-Poisson equation of order $s$, i.e. $\left( - \Delta_p\right)^s u = f$. In the range $p>1$ and $s\in \big(\frac{p-1}{p},1\big)$ we prove Calder\'on & Zygmund type estimates at the…

We present pointwise gradient bounds for solutions to $p$-Laplacean type non-homogeneous equations employing non-linear Wolff type potentials, and then prove similar bounds, via suitable caloric potentials, for solutions to parabolic…

We prove the unique solvability of second order elliptic equations in non-divergence form in Sobolev spaces. The coefficients of the second order terms are measurable in one variable and VMO in other variables. From this result, we obtain…

In this paper, we generalize the $W^{2,p}$ interior estimates of fully nonlinear elliptic equations that were obtained by Caffarelli in [1]. The generalizations are carried out in two directions. One is that we relax the regularity…

We give sufficient conditions under which solutions of finite-difference schemes in the space variable for second order possibly degenerate parabolic and elliptic equations admit estimates of spatial derivatives up to any given order…

We investigate regularity and a priori estimates for Fokker-Planck and Hamilton-Jacobi equations with unbounded ingredients driven by the fractional Laplacian of order $s\in(1/2,1)$. As for Fokker-Planck equations, we establish…

For the $\bar\partial$-Neumann problem on a regular coordinate domain $\Omega\subset \C^{n+1}$, we prove $\epsilon$-subelliptic estimates for an index $\epsilon$ which is in some cases better than $\epsilon=\frac1{2m}$ ($m$ being the {\it…

In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations $(-\Delta)^\sigma u = u^p$ with an isolated singularity, where $\sg \in (0, 1)$ and $\frac{n}{n-2\sg} < p < \frac{n+2\sg}{n-2\sg}$. We…

We prove a priori estimates in $L_\infty$ for a class of quasilinear stochastic partial differential equations. The estimates are obtained independently of the ellipticity constant $\varepsilon$ and thus imply analogous estimates for…

We develop a Fredholm alternative for a fractional elliptic operator~$\mathcal{L}$ of mixed order built on the notion of fractional gradient. This operator constitutes the nonlocal extension of the classical second order elliptic operators…

The aim of this paper is to establish $W^2_p$ estimate for non-divergence form second-order elliptic equations with the oblique derivative boundary condition in domains with small Lipschitz constants. Our result generalizes those in [14,…

We consider nonlinear integro-differential equations, like the ones that arise from stochastic control problems with purely jump L\`evy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior…