Related papers: Weighted $L^2$ Estimates for Elliptic Homogenizati…

We develop a new approach to the invertibility of the layer potentials on $L^p$ associated with elliptic equations and systems in Lipschitz domains. As a consequence, for $n\ge 4$ and $(2(n-1)/(n+1))-\epsilon<p<2$, we obtain the solvability…

This paper is devoted to establishing global $W^{2, p}$ estimate for strong solutions to the Dirichlet problem of uniformly elliptic equations in the non-divergence form where the domain is a Lipschitz polyhedra.

We establish interior Lipschitz estimates at the macroscopic scale for solutions to systems of linear elasticity with rapidly oscillating periodic coefficients and mixed boundary conditions in domains periodically perforated at a…

The main purpose of this work is to study uniform regularity estimates for a family of elliptic operators $\{\mathcal{L}_\varepsilon, \varepsilon>0\}$, arising in the theory of homogenization, with rapidly oscillating periodic coefficients.…

Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which…

We obtain the global weighted $W^{1,p}$ estimates for weak solutions of nonlinear elliptic equations over Reifenberg flat domains. Where nonlinearity $A(x,z,\xi)$ is assumed to be local uniform continuous in $z$ and have small BMO semi-norm…

We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in…

Let $\mathcal{L}_\epsilon$ be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients. We obtain the uniform $W^{1,p}$ estimate in a Lipschitz domain for solutions to the Dirichlet problem, where…

We study Neumann functions for divergence form, second order elliptic systems with bounded measurable coefficients in a bounded Lipschitz domain or a Lipschitz graph domain. We establish existence, uniqueness, and various estimates for the…

This paper presents an existence result and maximal regularity estimates for distributional solutions to degenerate/singular elliptic systems of $p$-Laplacian type with absorption and (prescribed) locally integrable forcing posed in…

We establish a global weighted $L^p$ estimate for the gradient of the solution to a divergence-form elliptic equations, where the coefficients are in a weighted VMO space and the equations have singularities on a co-dimension two boundary.

In a Lipschitz cylinder, this paper is devoted to establish an almost sharp error estimate $O(\varepsilon\log_2(1/\varepsilon))$ in $L^2$-norm for parabolic systems of elasticity with initial-Dirichlet conditions, arising in the…

This paper is concerned with a class of degenerate elliptic equations with rapidly oscillating coefficients in periodically perforated domains, which arises in the study of spectrum problems for uniformly elliptic equations in perforated…

We consider the linear elliptic systems or equations in divergence form with periodically oscillating coefficients. We prove the large-scale boundary Lipschitz estimate for the weak solutions in domains satisfying the so-called…

In this paper, we show that $W^{1,p}$ $(1\leq p<\infty)$ weak solutions to divergence form elliptic systems are Lipschitz and piecewise $C^{1}$ provided that the leading coefficients and data are of piecewise Dini mean oscillation, the…

We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with $L_2$ boundary data. The coefficients $A$ may depend on all variables, but are assumed to be close to…

We study rates of convergence of solutions in L^2 and H^{1/2} for a family of elliptic systems {L_\epsilon} with rapidly oscillating oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a…

In the present paper, we generalize the theory of quantitative homogenization for second-order elliptic systems with rapidly oscillating coefficients in $APW^2(\mathbb{R}^d)$, which is the space of almost-periodic functions in the sense of…

This article focuses on Lp-estimates for the square root of elliptic systems of second order in divergence form on a bounded domain. We treat complex bounded measurable coefficients and allow for mixed Dirichlet/Neumann boundary conditions…

We consider a generalized steady Stokes system with discontinuous coefficients in a nonsmooth domain when the inhomogeneous term belongs to a weighted $L^q$ space for $2<q<\infty$. We prove the global weighted $L^q$-estimates for the…