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We consider a class of fully nonlinear nonlocal degenerate elliptic operators which are modeled on the fractional Laplacian and converge to the truncated Laplacians. We investigate the validity of (strong) maximum and minimum principles,…

The aim of this paper is to adapt the notion of two-scale convergence in $L^p$ to the case of a measure converging to a singular one. We present a specific case when a thin cylinder with locally periodic rapidly oscillating boundary shrinks…

Let $\mathbb{H}^{n}$ be the Heisenberg group. For $0 \leq \alpha < Q=2n+2$ and $N \in \mathbb{N}$ we consider exponent functions $p(\cdot) : \mathbb{H}^{n} \to (0, +\infty)$, which satisfies H\"older conditions, such that $\frac{Q}{Q+N} <…

Here, we study quantitative homogenization of first-order convex Hamilton-Jacobi equations with $(u/\varepsilon)$-periodic Hamiltonians which typically appear in dislocation dynamics. Firstly, we establish the optimal convergence rate by…

We establish a necessary and sufficient criterion for the Fredholmness of a general locally compact band-dominated operator $A$ on $L^p(R)$ and solve the long-standing problem of computing its Fredholm index in terms of the limit operators…

We consider a general second order matrix operator in a multi-dimensional domain subject to a classical boundary condition. This operator is perturbed by a first order differential operator, the coefficients of which depend arbitrarily on a…

We study the Zariski closure of points in local deformation rings corresponding to potential semi-stable representations with certain prescribed $p$-adic Hodge theoretic properties. We show in favourable cases that the closure is equal to a…

We show how nonlocal boundary conditions of Robin type can be encoded in the pointwise expression of the fractional operator. Notably, the fractional Laplacian of functions satisfying homogeneous nonlocal Neumann conditions can be expressed…

In this paper, we deal with the family of Steklov sampling operators in the general setting of Orlicz spaces. The main result of the paper is a modular convergence theorem established following a density approach. To do this, a Luxemburg…

Local solvability and non-solvability are classified for left-invariant differential operators on the Heisenberg group H_1 of the form L=P_n(X,Y)+Q(X,Y) where the P_n are certain homogeneous polynomials of order n greater than or equal to 2…

We study the curl-div-system with variable coefficients and a nonlocal homogenisation problem associated with it. Using, in part refining, techniques from nonlocal $H$-convergence for closed Hilbert complexes, we define the appropriate…

Let $L$ be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and H\"older's continuity. Also assume that $L$ has a bounded holomorphic functional calculus on $L^2(\mathbb{R}^n)$. In this paper, we construct…

In this work we study the homogenization for eigenvalues of the fractional $p-$Laplace in a bounded domain both with Dirichlet and Neumann conditions. We obtain the convergence of eigenvalues and the explicit order of the convergence rates.

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…

We consider an eigenvalue problem for a divergence form elliptic operator $A_\epsilon$ with high contrast periodic coefficients with period $\epsilon$ in each coordinate, where $\epsilon$ is a small parameter. The coefficients are perturbed…

The periodic unfolding method is one of the latest tool after multi-scale convergence to study multi-scale problems like homogenization problems. It provides a good understanding of various micro scales involved in the problem which can be…

The operator $T$, defined by convolution with the affine arc length measure on the moment curve parametrized by $h(t)=(t,t^{2},...,t^{d})$ is a bounded operator from $L^{p}$ to $L^{q}$ if $(\frac{1}{p}, \frac{1}{q})$ lies on a line segment.…

We prove some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of fractional elliptic phases, according to…

We consider linear and obstacle problems driven by a nonlocal integral operator, for which nonlocal interactions are restricted to a ball of finite radius. These type of operators are used to model anomalous diffusion and, for a special…

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…