Related papers: Global W^1,p regularity for an elliptic problem wi…

The theory of second order complex coefficient operators of the form $\mathcal{L}=\mbox{div} A(x)\nabla$ has recently been developed under the assumption of $p$-ellipticity. In particular, if the matrix $A$ is $p$-elliptic, the solutions…

We study the divergence form second-order elliptic equations with mixed Dirichlet-conormal boundary conditions. The unique $W^{1,p}$ solvability is obtained with $p$ being in the optimal range $(4/3,4)$. The leading coefficients are assumed…

Using the variational approach and the critical point theory, we established several criteria for the existence of at least one nontrivial solution for a discrete elliptic boundary value problem with a weight $p(\cdot, \cdot)$ and depending…

In this paper, we establish global $C^{1,\alpha}$-regularity for bounded generalized solutions of elliptic equations in divergence form with Musielak-Orlicz growth and subject to Dirichlet or Neumann boundary conditions. In fact, our…

We consider degenerate and singular parabolic equations with $p$-Laplacian structure in bounded nonsmooth domains when the right-hand side is a signed Radon measure with finite total mass. We develop a new tool that allows global regularity…

We prove a global version of the classical result that $p$-harmonic functions belong to $W^{2,2}_{loc}$ for $1<p<3+\frac{2}{n-2}$. The proof relies on Cordes' matrix inequalities [7] and techniques from the work of Cianchi and Maz'ya [5,6].

In this paper, we continue the study of a class of second order elliptic operators of the form $\mathcal L=\mbox{div}(A\nabla\cdot)$ in a domain above a Lipschitz graph in $\mathbb R^n,$ where the coefficients of the matrix $A$ satisfy a…

We study a singular elliptic problem driven by a mixed local-nonlocal operator of the form \begin{equation*} \begin{aligned} -\Delta_p u + (-\Delta_q)^s u &= \frac{\lambda}{u^{\delta}} + u^r \text{ in } \Omega\newline u > 0 \text{ in }…

We derive global estimates in critical scale invariant norms for solutions of elliptic systems with antisymmetric potentials and almost holomorphic Hopf differential in two dimensions. Moreover we obtain new energy identities in such norms…

We prove global gradient estimates for parabolic $p$-Laplace type equations with measure data, whose model is $$u_t - \textrm{div} \left(|Du|^{p-2} Du\right) = \mu \quad \textrm{in} \ \Omega \times (0,T) \subset \mathbb{R}^n \times…

We study integrals over Hermitian supermatrices of arbitrary size $p+q$, that are parametrized by an external field $X$ and a source $Y$, of respective size $m+n$ and $p+q$. We show that these integrals exhibit a simple topological…

The purpose of this paper is to investigate the existence of three different weak solutions to a nonlinear elliptic problem that is governed by the weighted {\varphi}-Laplacian operator and subjected to Dirichlet boundary conditions. We…

We give a quantitative characterization of the pairs of weights $(w,v)$ for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak $(p,p)$ type inequality, for $1\leq p<\infty$. More…

This article presents new parabolic and elliptic type gradient estimates for positive smooth solutions to a nonlinear parabolic equation involving the Witten Laplacian in the context of smooth metric measure spaces. The metric and potential…

We show continuity in generalized weighted Morrey spaces of sub-linear integral operators generated by some classical integral operators and commutators. The obtained estimates are used to study global regularity of the solution of the…

We prove a Carleman estimate for elliptic second order partial differential operators with Lipschitz continuous coefficients. The Carleman estimate is valid for any complex-valued function $u\in W^{2,2}$ with support in a punctured ball of…

Elliptic and parabolic integro-differential model problems are considered in the whole space. By verifying H\"ormander condition, the existence and uniqueness is proved in L_{p}-spaces of functions whose regularity is defined by a scalable,…

Let $n\ge2$ and $\Omega\subset\mathbb{R}^n$ be a bounded NTA domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second order elliptic equations of divergence form…

We consider a non-local boundary value problem for the Laplace equation in unbounded studding the weak and strong solvability of that problem in the framework of the weighted Sobolev space $W^{1,p}_\nu$, with a Muckenhoupt weight. We proved…

In this paper, we provide suitable characterisations of pairs of weights $(V,W),$ known as Bessel pairs, that ensure the validity of weighted Hardy-type inequalities. The abstract approach adopted here makes it possible to establish such…