Related papers: Error estimate of the second-order homogenization …

The Riccati equation method is used to establish oscillation and non-oscillation criteria for second order linear nonhomogeneous functional-differential equations.We show that the obtained oscillation criterion is a generalization of J. S.…

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

We establish the Caccioppoli inequality, a reverse H\"older inequality in the spirit of the classic estimate of Meyers, and construct the fundamental solution for linear elliptic differential equations of order $2m$ with certain lower order…

We consider a general linear parabolic problem with extended time boundary conditions (including initial value problems and periodic ones), and approximate it by the implicit Euler scheme in time and the Gradient Discretisation method in…

A class of linear parabolic equations are considered. We derive a common framework for the a posteriori error analysis of certain second-order time discretisations combined with finite element discretisations in space. In particular we…

We study the behavior of second-order degenerate elliptic systems in divergence form with random coefficients which are stationary and ergodic. Assuming moment bounds like Chiarini and Deuschel [Arxiv preprint 1410.4483, 2014] on the…

We consider the problem of obtaining higher order in regularization parameter $\epsilon$ analytical results for master integrals with elliptics. The two commonly employed methods are provided by the use of differential equations and direct…

We study and characterize the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. We obtain that the optimal rate of convergence is either $O(\varepsilon)$ or $O(\varepsilon^2)$…

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…

In a Lipschitz cylinder, this paper is devoted to establish an almost sharp error estimate $O(\varepsilon\log_2(1/\varepsilon))$ in $L^2$-norm for parabolic systems of elasticity with initial-Dirichlet conditions, arising in the…

In this paper we consider nonlinear problems with an operator depending only on the deformation tensor. We consider the class of operators derived from a potential and with $(p,\delta)$ structure, for $1<p\leq 2$ and for all $\delta\geq0$.…

We analyze a divergence based first order system least squares method applied to a second order elliptic model problem with homogeneous boundary conditions. We prove optimal convergence in the $L^2(\Omega)$ norm for the scalar variable.…

In this study, we investigate the convergence rates for the homogenization of elliptic equations with lower-order terms under the spectral gap assumption, in both bounded domains and the entire space. Our analysis demonstrates that…

This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-\operatorname{div} A \nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces,…

We derive and analyze high order discontinuous Galerkin methods for second-order elliptic problems on implicitely defined surfaces in $\mathbb{R}^{3}$. This is done by carefully adapting the unified discontinuous Galerkin framework of…

A central question in numerical homogenization of partial differential equations with multiscale coefficients is the accurate computation of effective quantities, such as the homogenized coefficients. Computing homogenized coefficients…

For a nonlinear ordinary differential equation with time delay, the differentiation of the solution with respect to the delay is investigated. Special emphasis is laid on the second-order derivative. The results are applied to an associated…

We study partial H\"older regularity for nonlinear elliptic systems in divergence form with double-phase growth, modeling double-phase non-Newtonian fluids in the stationary case.

Lie's linearizability criteria for scalar second-order ordinary differential equations had been extended to systems of second-order ordinary differential equations by using geometric methods. These methods not only yield the linearizing…

We prove a self-improving property for reverse H{\"o}lder inequalities with non-local right hand side. We attempt to cover all the most important situations that one encounters when studying elliptic and parabolic partial differential…