Related papers: Compactness and Large-Scale Regularity for Darcy's…

We investigate quantitative estimates in periodic homogenization of second-order elliptic systems of elasticity with singular fourth-order perturbations. The convergence rates, which depend on the scale $\kappa$ that represents the strength…

We consider the steady Stokes equations supplemented with Navier boundary conditions including a non-negative friction coefficient. We prove maximal regularity estimates (including the prominent spaces $W^{1,p}$ and $W^{2,p}$ for…

Convergence results for the immersed boundary method applied to a model Stokes problem with the homogeneous Dirichlet boundary condition are presented. As a discretization method, we deal with the finite element method. First, the immersed…

We consider immersions admitting uniform representations as an L-Lipschitz graph. In codimension 1, we show compactness for such immersions for arbitrary fixed finite L and uniformly bounded volume. The same result is shown in arbitrary…

We study stationary Stokes systems in divergence form with piecewise Dini mean oscillation coefficients and data in a bounded domain containing a finite number of subdomains with $C^{1,\rm{Dini}}$ boundaries. We prove that if $(u, p)$ is a…

We consider the homogenization of the Poisson and the Stokes equations in the whole space perforated with periodically distributed small holes. The periodic homogenization in bounded domains is well understood, following the classical…

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…

We prove localized energy estimates for the wave equation in domains with a strictly concave boundary when homogeneous Dirichlet or Neumann conditions are imposed. By restricting the solution to small, frequency dependent, space time…

We prove the $L_1$ in time and $B^{s+1}_{q,1}\times B^s_{q,1}$ in space maximal regularity for the Stokes equations in the viscous compressible fluid flows in domains in the $N$ dimensional Euclidean space $R^N$ whose boundary is $C^3$…

We develop a quantitative theory of stochastic homogenization for linear, uniformly parabolic equations with coefficients depending on space and time. Inspired by recent works in the elliptic setting, our analysis is focused on certain…

We prove some $L^{\infty}$ a priori estimates as well as existence and stability theorems for the weak solutions of the complex Hessian equations in domains of $C^n$ and on compact K\"ahler manifolds. We also show optimal $L^p$…

In this paper we study the Dirichlet problem for Laplace's equation in a domain $\omega_{\epsilon, \eta}$ perforated periodically with small holes in $\mathbb{R}^d$, where $\epsilon$ represents the scale of the minimal distances between…

In this paper we analyze the finite element approximation of the Stokes equations with non-smooth Dirichlet boundary data. To define the discrete solution, we first approximate the boundary datum by a smooth one and then apply a standard…

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…

This paper is devoted to the maximal $L^1$ regularity and asymptotic behavior for solutions to the inhomogeneous incompressible Navier-Stokes equations under a scaling-invariant smallness assumption on the initial velocity. We obtain a new…

This paper is concerned with boundary regularity estimates in the homogenization of elliptic equations with rapidly oscillating and high-contrast coefficients. We establish uniform nontangential-maximal-function estimates for the Dirichlet,…

We prove generalized Gaffney inequalities and the discrete compactness for finite element differential forms on $s$-regular domains, including general Lipschitz domains. In computational electromagnetism, special cases of these results have…

This paper studies the convergence rates in $L^2$ and $H^1$ of Dirichelt problems for Stokes systems with rapidly oscillating periodic coefficients, without any regularity assumptions on the coefficients.

In this paper, we study the regularity for viscosity solutions of locally uniformly elliptic equations and obtain a series of interior pointwise $C^{k,\alpha}$ ($k\geq 1$, $0<\alpha<1$) regularity with smallness assumptions on the solution…

In this paper, we mainly employed the idea of the previous paper to study the sharp uniform $W^{1,p}$ estimates with $1<p\leq \infty$ for more general elliptic systems with the Neumann boundary condition on a bounded $C^{1,\eta}$ domain,…