Related papers: Homogenization of generalized second-order ellipti…

In this work we study the homogenization problem for nonlinear eigenvalues of quasilinear elliptic operators. We obtain an explicit order of convergence in $k$ and in $\varepsilon$ for the (variational) eigenvalues.

Suppose that ${\cal L}$ is a divergence form differential operator of the form ${\cal L}f:=(1/2) e^{U}\nabla_x\cdot\big[e^{-U}(I+H)\nabla_x f\big]$, where $U$ is scalar valued, $I$ identity matrix and $H$ an anti-symmetric matrix valued…

A general analysis framework is presented in this paper for many different types of finite element methods (including various discontinuous Galerkin methods). For second order elliptic equation, this framework employs $4$ different…

The paper deals with the second order regularity properties of the weak solutions $u\in W^{1,\phi}(\Omega, \real^n)$ } of systems of the form \begin{equation*}\label{equareg} -\dive A(x,\E u)=f, \end{equation*} in a bounded domain…

In this note we describe how the Neumann homogenization of fully nonlinear elliptic equations can be recast as the study of nonlocal (integro-differential) equations involving elliptic integro-differential operators on the boundary. This is…

Let $L$ be a second-order elliptic operator with analytic coefficients defined in $B_1\subseteq\mathbb R^n$. We construct explicitly and canonically a fundamental solution for the operator, i.e., a function $u:B_{r_0}\to\mathbb R$ such that…

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…

In this work, we generalize Sacks-Uhlenbeck's existence result for harmonic spheres, constructing for $n \ge 2$, regular, non-trivial, $n$-harmonic $n$-spheres into suitable target manifolds. We obtain an infinite family of new…

We give a simplified presentation of the obstacle problem approach to stochastic homogenization for elliptic equations in nondivergence form. Our argument also applies to equations which depend on the gradient of the unknown function. In…

We study the approximation of the spectrum of a second-order elliptic differential operator by the Hybrid High-Order (HHO) method. The HHO method is formulated using cell and face unknowns which are polynomials of some degree $k\geq0$. The…

Guaranteed upper-lower bounds on homogenized coefficients, arising from the periodic cell problem, are calculated in a scalar elliptic setting. Our approach builds on the recent variational reformulation of the Moulinec-Suquet (1994) Fast…

For two scale elliptic equations in a domain $D$, standard homogenization errors are deduced with the assumption that the solution $u_0$ of the homogenized equation belongs to $H^2(D)$. For two scale Maxwell equations, the corresponding…

The renewed interest in investigating quaternionic quantum mechanics, in particular tunneling effects, and the recent results on quaternionic differential operators motivate the study of resolution methods for quaternionic differential…

We investigate Lawruk elliptic boundary-value problems for homogeneous differential equations in a two-sided refined Sobolev scale. These problems contain additional unknown functions in the boundary conditions of arbitrary orders. The…

This paper deals with generalized differentiability and second-order necessary optimality conditions for a box-constrained optimal control problem governed by an exponential semilinear elliptic equation with discrete measures as sources,…

In this paper we study a second order dynamical system with variable coefficients in connection to the minimization problem of a smooth nonconvex function. The convergence of the trajectories generated by the dynamical system to a critical…

We consider a non-uniformly elliptic second-order differential operator with periodic coefficients that models composite media consisting of highly anisotropic cylindrical fibres periodically distributed in an isotropic background. The…

Aim of this paper is to prove the second order differentiation formula along geodesics in compact $RCD^*(K,N)$ spaces with $N<\infty$. This formula is new even in the context of Alexandrov spaces. We establish this result by showing that…

We study elliptic gradient systems with fractional laplacian operators on the whole space $$ (- \Delta)^\mathbf s \mathbf u =\nabla H (\mathbf u) \ \ \text{in}\ \ \mathbf{R}^n,$$ where $\mathbf u:\mathbf{R}^n\to \mathbf{R}^m$, $H\in…

Let $\mathcal{O}\subset\mathbb{R}^d$ a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a self-adjoint matrix strongly elliptic second order differential operator $B_{D,\varepsilon}$, $0<\varepsilon…