Related papers: Note on quantitative homogenization results for pa…

We consider the parabolic type equation in $\mathbb{R}^n$: \begin{align}\label{equ-0} (\partial_t+H)y(t,x)=0,\,\,\, (t,x)\in (0,\infty)\times\mathbb{R}^n;\;\; \quad y(0,x)\in L^2(\mathbb{R}^n), \end{align} where $H$ can be one of the…

The paper deals with periodic homogenization of nonlocal symmetric convolution type operators in $L^2(\mathbb R^d)$, whose kernel is the product of a density that belongs to the domain of attraction of an $\alpha$-stable law and a rapidly…

The homogenization of elliptic divergence-type fourth-order operators with periodic coefficients is studied in a (periodic) domain. The aim is to find an operator with constant coefficients and represent the equation through a perturbation…

For a general class of difference operators $H_\varepsilon = T_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon\mathbb{Z}^d)$, where $V_\varepsilon$ is a multi-well potential and $\varepsilon$ is a small parameter, we analyze the…

We consider the homogeneous equation ${\mathcal A} u=0$, where ${\mathcal A}$ is a symmetric and coercive elliptic operator in $H^1(\Omega)$ with $\Omega$ bounded domain in ${{\mathbb R}}^d$. The boundary conditions involve fractional power…

We prove a sharp Weyl estimate for the number of eigenvalues belonging to a fixed interval of energy of a self-adjoint difference operator acting on $\ell^2(\epsilon\mathbb{Z}^d)$ if the associated symplectic volume of phase space in…

We use methods of harmonic analysis and group representation theory to study the spectral properties of the abstract parabolic operator $\mathscr L = -d/dt+A$ in homogeneous function spaces. We provide sufficient conditions for…

We are interested in the homogenization of elastic-electric coupling equation, with rapidly oscillating coefficients, in periodically perforated piezoelectric body. We justify the two first terms in the usual asymptotic development of the…

We consider an arbitrary metric graph, to which we glue another graph with edges of lengths proportional to $\varepsilon$, where $\varepsilon$ is a small positive parameter. On such graph, we consider a general self-adjoint second order…

Given a complex, elliptic coefficient function we investigate for which values of $p$ the corresponding second-order divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly…

Let $\mathcal{X}$ be a real separable Hilbert space. Let $C$ be a linear, bounded and positive operator on $\mathcal{X}$ and let $A$ be the infinitesimal generator of a strongly continuous semigroup on $\mathcal{X}$. Let $\{W(t)\}_{t\geq…

We consider operators of the form ${\mathcal L}=-L-V$, where $L$ is an elliptic operator and $V$ is a singular potential, defined on a smooth bounded domain $\Omega\subset \R^n$ with Dirichlet boundary conditions. We allow the boundary of…

We prove a parabolic analogue of Wolff's inequality adapted to the intrinsic scaling $\delta_c(x,t)=(cx,c^2t)$ and formulated in terms of time-backward parabolic dyadic rectangles. As a consequence, we obtain equivalent characterizations of…

In this paper, we study the estimates of resolvents $ R(\lambda,\mathcal{L}_{\varepsilon})=(\mathcal{L}_{\varepsilon}-\lambda I)^{-1} $, where $$ \mathcal{L}_{\varepsilon}=-\operatorname{div}(A(x/\varepsilon)\nabla) $$ is a family of second…

We consider an iteration method for solving an elliptic type boundary value problem $\mathcal{A} u=f$, where a positive definite operator $\mathcal{A}$ is generated by a quasi--periodic structure with rapidly changing coefficients (typical…

In this work we consider the identifiability of two coefficients $a(u)$ and $c(x)$ in a quasilinear elliptic partial differential equation from observation of the Dirichlet-to-Neumann map. We use a linearization procedure due to Isakov [On…

We construct the field theory which describes the universal properties of the quasi-static isotropic depinning transition for interfaces and elastic periodic systems at zero temperature, taking properly into account the non-analytic form of…

In his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order evolution equation which displays some challenging features. In the opening of his 1967 hypoellipticity paper H\"ormander discussed a…

This paper investigates homogenization problems for the nonlocal operators with rapidly oscillating coefficients in the cases of periodic and random statistically homogeneous micro-structures. These operators involve the fractional…

This paper is the companion article to [Ann. Probab. 39 (2011) 779--856]. We consider a discrete elliptic equation on the $d$-dimensional lattice $\mathbb{Z}^d$ with random coefficients $A$ of the simplest type: They are identically…