arXiv subjects

Nikita N. Senik

Publications and source records attributed to Nikita N. Senik.

3 recordsLinked to original sources

arXiv2021

Homogenization for locally periodic elliptic problems on a domain

Let $Ω$ 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 $Ω$. We suppose that $\varepsilon$ is small and the function $A$ is Lipschitz in the first variable and periodic in the second, so the coefficients of $\mathcal A^\varepsilon$ are locally periodic and rapidly oscillate. Given $μ$ in the resolvent set, we are interested in finding the rates of approximations, as $\varepsilon\to0$, for $(\mathcal A^\varepsilon-μ)^{-1}$ and $\nabla(\mathcal A^\varepsilon-μ)^{-1}$ in the operator topology on $L_p$ for suitable $p$. It is well-known that the rates depend on regularity of the effective operator $\mathcal A^0$. We prove that if $(\mathcal A^0-μ)^{-1}$ and its adjoint are bounded from $L_p(Ω)^n$ to the Lipschitz--Besov space $Λ_p^{1+s}(Ω)^n$ with $s\in(0,1]$, then the rates are, respectively, $\varepsilon^s$ and $\varepsilon^{s/p}$. The results are applied to the Dirichlet, Neumann and mixed Dirichlet--Neumann problems for strongly elliptic operators with uniformly bounded and $\operatorname{VMO}$ coefficients.

math.AP↗

arXiv2017

Homogenization for non-self-adjoint locally periodic elliptic operators

We study the homogenization problem for matrix strongly elliptic operators on $L_2(\mathbb R^d)^n$ of the form $\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$. The function $A$ is Lipschitz in the first variable and periodic in the second. We do not require that $A^*=A$, so $\mathcal A^\varepsilon$ need not be self-adjoint. In this paper, we provide, for small $\varepsilon$, two terms in the uniform approximation for $(\mathcal A^\varepsilon-μ)^{-1}$ and a first term in the uniform approximation for $\nabla(\mathcal A^\varepsilon-μ)^{-1}$. Primary attention is paid to proving sharp-order bounds on the errors of the approximations.

math.AP↗

arXiv2015

Homogenization for non-self-adjoint periodic elliptic operators on an infinite cylinder

We consider the problem of homogenization for non-self-adjoint second-order elliptic differential operators~$\mathcal{A}^{\varepsilon}$ of divergence form on $L_{2}(\mathbb{R}^{d_{1}}\times\mathbb{T}^{d_{2}})$, where $d_{1}$ is positive and~$d_{2}$ is non-negative. The~coefficients of the operator~$\mathcal{A}^{\varepsilon}$ are periodic in the first variable with period~$\varepsilon$ and smooth in a certain sense in the second. We show that, as $\varepsilon$ gets small, $(\mathcal{A}^{\varepsilon}-μ)^{-1}$ and~$D_{x_{2}}(\mathcal{A}^{\varepsilon}-μ)^{-1}$ converge in the operator norm to, respectively, $(\mathcal{A}^{0}-μ)^{-1}$ and~$D_{x_{2}}(\mathcal{A}^{0}-μ)^{-1}$, where $\mathcal{A}^{0}$ is an operator whose coefficients depend only on~$x_{2}$. We also obtain an approximation for $D_{x_{1}}(\mathcal{A}^{\varepsilon}-μ)^{-1}$ and find the next term in the approximation for~$(\mathcal{A}^{\varepsilon}-μ)^{-1}$. Estimates for the rates of convergence and the rates of approximation are provided and are sharp with respect to the order.

math.AP↗