Related papers: Higher Order Elliptic Equations on Nonsmooth Domai…
This paper was devoted to study the quantitative homogenization problems for nonlinear elliptic operators in perforated domains. We obtained a sharp error estimate $O(\varepsilon)$ when the problem was anchored in the reference domain…
This paper is concerned with the homogenization of the Dirichlet eigenvalue problem, posed in a bounded domain $\Omega\subset\mathbb R^2$, for a vectorial elliptic operator $-\nabla\cdot A^\epsilon(\cdot)\nabla$ with $\epsilon$-periodic…
In this paper, we investigate the Dirichlet problem on lower dimensional manifolds for a class of weighted elliptic equations with coefficients that are singular on such sets. Specifically, we study the problem \[\begin{cases} -{\rm…
Concerned with the Stokes systems with rapidly oscillating periodic coefficients, we mainly extend the recent works in \cite{SGZWS,G} to those in term of Lipschitz domains. The arguments employed here are quite different from theirs, and…
A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic…
This paper is concerned with the large-scale regularity in the homogenization of elliptic systems of elasticity with periodic high-contrast coefficients. We obtain the large-scale Lipschitz estimate that is uniform with respect to the…
We consider elliptic equations and systems in divergence form with the conormal or the Robin boundary conditions, with small BMO (bounded mean oscillation) or variably partially small BMO coefficients. We propose a new class of domains…
We study H\"older continuity of solutions to the Dirichlet problem for measures having density in $L^p$, $p>1$, with respect to Hausdorff-Riesz measures of order $2n-2+\epsilon$ for $0<\epsilon \leq 2$, in a bounded strongly hyperconvex…
We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale $L^\infty$-type estimate for the gradient of a solution. The estimate…
This paper examines the solvability of the equation $\mathrm{div} \ \mathbf{u} = f$ with a zero Dirichlet boundary condition for $\mathbf{u}$. A classical result establishes that for a bounded domain $\Omega \subset \mathbb{R}^N$ with a…
This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain $\Omega\subset \reals^n$, $n\geq2$, with boundary that is decomposed as $\partial\Omega=D\cup N$, $D$ and $N$ disjoint. We let…
In this paper, we study the existence of a solution for a class of Dirichlet problems with a singularity and a convection term. Precisely, we consider the existence of a positive solution to the Dirichlet problem $$-\Delta_p u =…
In this paper, we mainly establish the existence of at least three non-trivial solutions for a class of nonhomogeneous quasilinear elliptic systems with Dirichlet boundary value or Neumann boundary value in a bounded domain…
We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on $\mathbb{R}^d$ with stationary law (i.e. spatially…
In this paper we investigate elliptic partial differential equations on Lipschitz domains in the plane whose coefficient matrices have small (but possibly nonzero) imaginary parts and depend only on one of the two coordinates. We show that…
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…
In this paper, we prove sharp estimates and existence results for anisotropic nonlinear elliptic problems with lower order terms depending on the gradient. Our prototype is: $ \left\{ \begin{array}{ll} -\mathcal Q_{p}u =[H(Du)]^{q}+f(x)…
In this paper, we are interested in the reiterated homogenization of linear elliptic equations of the form $-\frac{\partial}{\partial x_{i}} \left(a_{i j} \left(\frac{x}{\varepsilon}, \frac{x}{\varepsilon^{2}}\right) \frac{\partial…
We study the optimal approximation of the solution of an operator equation by certain n-term approximations with respect to specific classes of frames. We study worst case errors and the optimal order of convergence and define suitable…
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…