Related papers: On homogenization of the first initial-boundary va…
We obtained estimates for first eigenvalues of the Dirichlet boundary value problem for elliptic operators in divergence form (i.e. for the principal frequency of non-homogeneous membranes) in bounded domains $\Omega \subset \mathbb C$…
For a family of elliptic operators with periodically oscillating coefficients, $-\text{div}( A(\cdot/\varepsilon) \nabla) $ with tiny $\varepsilon>0$, we comprehensively study the first-order expansions of eigenvalues and eigenfunctions…
We study well-posedness of boundary value problems of Dirichlet and Neumann type for elliptic systems on the upper half-space with coefficients independent of the transversal variable, and with boundary data in fractional…
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…
We consider the initial boundary value problem for the time-fractional diffusion equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial data $a(x)\in L^{2}(D)$ in a bounded domain $D\subset \mathbb{R}^d$ with…
Hyperbolic systems of the first and higher-order partial differential equations appear in many multiphysics problems. We will be dealing with a wave propagation problem in a piece-wise homogeneous medium. Mathematically, the problem is…
Homogenization for non-local operators in periodic environments has been studied intensively. So far, these works are mainly devoted to the qualitative results, that is, to determine explicitly the operators in the limit. To the best of…
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 $L_2({\mathbb R}^d; {\mathbb C}^n)$, 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})^*…
We prove the existence of unique solutions to the Dirichlet boundary value problems for linear second-order uniformly parabolic operators in either divergence or non-divergence form with boundary blowup low-order coefficients. The domain is…
In the whole space $R^d$, $d\ge 2$, we study homogenization of a divergence form elliptic fourth-order operator $A_\varepsilon$ with measurable $\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter. For the resolvent…
In the whole space $R^d$, $d\ge 2$, we study homogenization of a divergence form elliptic operator $A_\varepsilon$ of order $2m\ge 4$ with measurable $\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter. For the…
In $L_2(\mathbb{R}^d;{\mathbb C}^n)$ we consider selfadjoint strongly elliptic second order differential operators ${\mathcal A}_\varepsilon$ with periodic coefficients depending on ${\mathbf x}/\varepsilon$. We study the behavior of the…
Consider the linear parabolic operator in divergence form $$\mathcal{H} u =\partial_t u(X,t)-\text{div}(A(X)\nabla u(X,t)).$$ We employ a method of Dahlberg to show that the Dirichlet problem for $\mathcal{H}$ in the upper half plane is…
We study second-order hyperbolic equations with degenerate elliptic operators and non-homogeneous Dirichlet boundary inputs. We establish existence and regularity of weak solutions in weighted Sobolev spaces under mild assumptions on the…
We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence type elliptic operators. The construction is applied in two settings. First, we show…
In this paper, we focus on the homogenization process of the non-local elliptic boundary value problem $$\mathcal{L}_\varepsilon^s u_\varepsilon =(-\nabla\cdot (A_\varepsilon(x)\nabla))^{s}u_\varepsilon=f \mbox{ in } \mathcal O, $$ with…
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…
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…
The paper deals with periodic homogenization problem for a para\-bo\-lic equation whose elliptic part is a convolution type operator with rapidly oscillating coefficients. It is assumed that the coefficients are rapidly oscillating periodic…