Related papers: The L^p Dirichlet problem for second-order, non-di…

We consider the mixed problem for the Laplace operator in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. The boundary of the domain is decomposed into two disjoint sets D and N. We suppose the…

A second-order regularity theory is developed for solutions to a class of quasilinear elliptic equations in divergence form, including the $p$-Laplace equation, with merely square-integrable right-hand side. Our results amount to the…

We find necessary and sufficient conditions for the validity of weighted Rellich and Calderon-Zygmund inequalities in L^p, 1 \leq p \leq \infty, in the whole space and in the half-space with Dirichlet boundary conditions. General operators…

This is the third part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. For $L$ in some class of elliptic operators, we study weighted norm $L^p$ inequalities for singular…

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…

This study investigates Dirichlet boundary condition related to a class of nonlinear parabolic problem with nonnegative $L^1$-data, which has a variable-order fractional $p$-Laplacian operator. The existence and uniqueness of renormalized…

We show the $L^r(\mathbb{R}^d, \mu)$-uniqueness for any $r \in (1, 2]$ and the essential self-adjointness of a Dirichlet operator $Lf = \Delta f +\langle \frac{1}{\rho}\nabla \rho , \nabla f \rangle$, $f \in C_0^{\infty}(\mathbb{R}^d)$ with…

In this paper, we investigate the boundary behavior of solutions of divergence-form operators with an elliptic symmetric part and a $BMO$ anti-symmetric part. Our results will hold in non-tangentially accessible (NTA) domains; these general…

Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we study a selfadjoint matrix elliptic second order differential operator $B_{D,\varepsilon}$, $0<\varepsilon\leqslant 1$, with…

We obtain $L^p(L^q)$ maximal regularity estimates for time dependent second order elliptic operators in divergence form with rough dependencies in the spatial variables.

This article offers a study of the Calder\'on type inverse problem of determining up to second order coefficients of the higher order elliptic operator. Here we show that it is possible to determine an anisotropic second order perturbation…

We consider the Dirichlet problems for second order linear elliptic equations in non-divergence and divergence forms on a bounded domain $\Omega$ in $\mathbb{R}^n$, $n \ge 2$: $$ -\sum_{i,j=1}^n a^{ij}D_{ij} u + b \cdot D u + cu = f…

We study some Dirichlet problem for a $p$--Laplacian type operator in the setting of Orlicz--Zygmund space $L^q\log^{-\alpha}L(\Omega,\mathbb R^N)$, $q >1$ and $\alpha>0$. More precisely, our aim is to establish which assuptions on the…

We investigate the validity and failure of Liouville theorems and Harnack inequalities for parabolic and elliptic operators with low regularity coefficients. We are particularly interested in operators of the form $\partial_t - \Delta…

We show existence and uniqueness for the solutions of the regularity and the Neumann problems for harmonic functions on Lipschitz domains with data in the Hardy spaces H^p, p>2/3, where This in turn implies that solutions to the Dirichlet…

We prove regularity estimates for weak solutions to the Dirichlet problem for a divergence form elliptic operator. We give $L^p$ estimates for the second derivative for $p<2$. Our work generalizes results due to Miranda [28].

In this paper we study perturbed Ornstein-Uhlenbeck operators \begin{align*}[\mathcal{L}_{\infty} v](x)=A\triangle v(x)+\langle Sx,\nabla v(x)\rangle-B v(x),\,x\in\mathbb{R}^d,\,d\geqslant 2,\end{align*} for simultaneously diagonalizable…

We generalize the Donsker-Varadhan minimax formula for the principal eigenvalue of a uniformly elliptic operator in nondivergence form to the first principal half-eigenvalue of a fully nonlinear operator which is concave (or convex) and…

We consider the operators \[ \nabla_X\cdot(A(X)\nabla_X),\ \nabla_X\cdot(A(X)\nabla_X)-\partial_t,\ \nabla_X\cdot(A(X)\nabla_X)+X\cdot\nabla_Y-\partial_t, \] where $X\in \Omega$, $(X,t)\in \Omega\times \mathbb R$ and $(X,Y,t)\in…

Let $A$ be a homogeneous elliptic differential operator of order $m$ on $% \Bbb{R}^{N}$ with constant complex coefficients. A partial version of the main result is as follows: Suppose that $u\in L_{loc}^{1}$ and that $Au\in L^{p}$ for some…