Related papers: The $L^p$ Dirichlet boundary problem for second or…

We study the solvability of boundary-value problems for differential-operator equations of the second order in L p (0, 1; X), with 1 < p < +$\infty$, X being a UMD complex Banach space. The originality of this work lies in the fact that we…

We provide a general construction scheme for $\mathcal L^p$-strong Feller processes on locally compact separable metric spaces. Starting from a regular Dirichlet form and specified regularity assumptions, we construct an associated…

In this paper we study boundary value problems for higher order elliptic differential operators in divergence form. We establish well posedness for problems with boundary data in Besov spaces $\dot B^{p,p}_s$, $p\leq 1$, given well…

We consider the Kato problem and extensions for degenerate elliptic operators of arbitrary order $2m$ ($m\geq 1$), whose coefficients are measurable, complex-valued and satisfy the G$\mathring{a}$rding inequality with respect to a…

We consider layer potentials associated to elliptic operators $Lu=-{\rm div}(A \nabla u)$ acting in the upper half-space $\mathbb{R}^{n+1}_+$ for $n\geq 2$, or more generally, in a Lipschitz graph domain, where the coefficient matrix $A$ is…

We establish uniform Lipschitz estimates for second-order elliptic systems in divergence form with rapidly oscillating, almost-periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded…

In this article we prove solvability results for $L^2$ boundary value problems of some elliptic systems $Lu=0$ on the upper half-space $\R^{n+1}_{+}, n\ge 1$, with transversally independent coefficients. We use the first order formalism…

We start presenting an $L^{\infty}$-gradient bound for solutions to non-homogeneous $p$-Laplacean type systems and equations, via suitable non-linear potentials of the right hand side. Such a bound implies a Lorentz space characterization…

We obtain the Kato square root estimate for second order elliptic operators in divergence form with mixed boundary conditions on an open and possibly unbounded set in $\mathbb{R}^d$ under two simple geometric conditions: The Dirichlet…

We use an iteration procedure propped up by a a classical form of the maximum principle to show the existence of solutions to a nonlinear Poisson equation with Dirichlet boundary conditions. These methods can be applied to the case of…

We prove that the $L^{p'}$-solvability of the homogeneous Dirichlet problem for an elliptic operator $L=-\operatorname{div}A\nabla$ with real and merely bounded coefficients is equivalent to the $L^{p'}$-solvability of the Poisson Dirichlet…

Given two elliptic operators L and M in nondivergence form, with coefficients A_L(x), A_M(x) and drift terms b_L(x), b_M(x), respectively, satisfying a Carleson measure disagreement condition in a Lipschitz domain Omega in R^{n+1}, then…

In this paper, we prove that there exists a unique, bounded continuous weak solution to the Dirichlet boundary value problem for a general class of second-order elliptic operators with singular coefficients, which does not necessarily have…

Let $n\ge2$, $\Omega\subset\mathbb{R}^n$ be a bounded one-sided chord arc domain, and $p\in(1,\infty)$. In this article, we study the (weak) $L^p$ Poisson--Robin(-regularity) problem for a uniformly elliptic operator…

In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $-\partial_tu + \mbox{div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under…

We study well-posedness of boundary value problems of Dirichlet and Neumann type for elliptic systems on the upper half-space with coefficients independent of the transversal variable, and with boundary data in fractional…

Second-order estimates are established for solutions to the $p$-Laplace system with right-hand side in $L^2$. The nonlinear expression of the gradient under the divergence operator is shown to belong to $W^{1,2}$, and hence to enjoy the…

This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^2$. Assuming that the principal coefficients satisfy the Dini mean oscillation condition, we establish the…

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega\subset\R^N$ with Dirichlet boundary conditions on $\partial\Omega$. The operator $L$ is a uniformly elliptic linear operator of…

We prove the first positive results concerning boundary value problems in the upper half-space of second order parabolic systems only assuming measurability and some transversal regularity in the coefficients of the elliptic part. To do so,…