Related papers: Operator error estimates for homogenization of the…

In $L_2({\mathbb R}^d; {\mathbb C}^n)$, we consider a matrix strongly elliptic differential operator ${A}_\varepsilon$ of order $2p$, $p \geqslant 2$. The operator ${A}_\varepsilon$ is given by ${A}_\varepsilon = b(\mathbf{D})^*…

Let $\Omega$ be a Lipschitz domain in $\mathbb R^n$ $n\geq 2,$ and $L=\mbox{div} (A\nabla\cdot)$ be a second order elliptic operator in divergence form. We establish solvability of the Dirichlet regularity problem with boundary data in…

In $L_2({\mathbb R}^d;{\mathbb C}^n)$, we consider a selfadjoint operator ${\mathcal B}_\varepsilon$, $0< \varepsilon \leqslant 1$, given by the differential expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon)b({\mathbf D}) +…

We consider a sequence of Dirichlet problems in varying domains (or, more generally, of relaxed Dirichlet problems involving measures in M_0) for second order linear elliptic operators in divergence form with varying matrices of…

Let $\Omega$ be a Lipschitz domain in $\mathbb R^d$, and let $\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$ be a strongly elliptic operator on $\Omega$. We suppose that $\varepsilon$ is small and the function $A$ is…

We study the regularity of solutions of elliptic second order boundary value problems on a bounded domain $\Omega$ in $\mathbb R^3$. The coefficients are not necessarily continuous and the boundary conditions may be mixed, i.e. Dirichlet on…

In this paper, we are interested in the periodic homogenization of quasilinear elliptic equations. We obtain error estimates $O(\varepsilon^{1/2})$ for a $C^{1,1}$ domain, and $O(\varepsilon^\sigma)$ for a Lipschitz domain, in which…

We consider a singularly perturbed Dirichlet spectral problem for an elliptic operator of second order. The coefficients of the operator are assumed to be locally periodic and oscillating in the scale $\varepsilon$. We describe the leading…

For a second order differential operator $A(\msx) =-\nabla a(\msx)\nabla + b'(\msx)\nabla+ \nabla \big(\msb''(\msx) \cdot\big)$ on a bounded domain $D$ with the Dirichlet boundary conditions on $\partial D$ there exists the inverse…

This work is about global H\"older regularity for solutions to elliptic partial differential equations subject to mixed boundary conditions on irregular domains. There are two main results. In the first, we show that if the domain of the…

We study spectral estimates of the divergence form uniform elliptic operators $-\textrm{div}[A(z) \nabla f(z)]$ with the Dirichlet boundary condition in bounded non-Lipschitz simply connected domains $\Omega \subset \mathbb C$. The…

We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. Under the Lipschitz condition on the coefficients we characterize the domain of the Poisson operators…

In this paper we study the Dirichlet problem for real-valued second order divergence form elliptic operators with boundary data in H\"{o}lder spaces. Our context is that of open sets $\Omega \subset \mathbb{R}^{n+1}$, $n \ge 2$, satisfying…

We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. For such a class of operators we establish a factorization into a product of first order operators…

Let $\Omega$ be a Lipschitz domain in $\mathbb R^n,n\geq 3,$ and $L=\divt A\nabla$ be a second order elliptic operator in divergence form. We will establish that the solvability of the Dirichlet regularity problem for boundary data in…

Homogenization of a scalar elliptic equation in a bounded domain with Neuman boundary condition is studied. Coefficients of the operator are oscillating over two different groups of variables with different small periods $\varepsilon$ and…

For a family of elliptic operators with rapidly oscillating periodic coefficients, we study the convergence rates for Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The results rely on an…

In $L_2 (\mathbb{R}^d; \mathbb{C}^n)$, we consider a selfadjoint matrix strongly elliptic second order differential operator $\mathcal{A}_\varepsilon$ with periodic coefficients depending on $\mathbf{x}/\varepsilon$. We find approximations…

We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic equation with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The…

The operator \[ A_{\varepsilon}= D_{1} g_{1}(x_{1}/\varepsilon, x_{2}) D_{1} + D_{2} g_{2}(x_{1}/\varepsilon, x_{2}) D_{2} \] is considered in $L_{2}({\mathbb{R}}^{2})$, where $g_{j}(x_{1},x_{2})$, $j=1,2,$ are periodic in $x_{1}$ with…