Related papers: Gradient estimates and the fundamental solution fo…
We present a framework that relates preconditioning with a posteriori error estimates in finite element methods. In particular, we use standard tools in subspace correction methods to obtain reliable and efficient error estimators. As a…
We will show that the same type of estimates known for the fundamental solutions for scalar parabolic equations with smooth enough coefficients hold for the first order derivatives of fundamental solution with respect to space variables of…
We consider an inverse problem for a higher order elliptic operator where the principal part is the polyharmonic operator $(-\Delta)^m$ with $ m \geq 2$. We show that the map from the coefficients to a certain bilinear form is injective. We…
In this paper we present a preliminary study on the Dirichlet-to-Neumann operator with respect to a second order elliptic operator with measurable coefficients, including first order terms, namely, the operator on $L^2(\partial\Omega)$…
We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address…
The purpose of this article is to formulate and prove a global high-order regularity result within the high-contrast framework of elliptic homogenization. In order to achieve this, we also present a version of the high-contrast Caccioppoli…
The theory of second order complex coefficient operators of the form $\mathcal{L}=\mbox{div} A(x)\nabla$ has recently been developed under the assumption of $p$-ellipticity. In particular, if the matrix $A$ is $p$-elliptic, the solutions…
We propose a multiscale approach for an elliptic multiscale setting with general unstructured diffusion coefficients that is able to achieve high-order convergence rates with respect to the mesh parameter and the polynomial degree. The…
In this paper, we prove sharp estimates and existence results for anisotropic nonlinear elliptic problems with lower order terms depending on the gradient. Our prototype is: $ \left\{ \begin{array}{ll} -\mathcal Q_{p}u =[H(Du)]^{q}+f(x)…
The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that…
Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give certain relations in the enveloping algebra,…
A typical system of k difference (or differential) equations can be compressed, or folded into a difference (or ordinary differential) equation of order k. Such foldings appear in control theory as the canonical forms of the controllability…
Differential equations on spaces of operators are very little developed in Mathematics, being in general very challenging. Here, we study a novel system of such (non-linear) differential equations. We show it has a unique solution for all…
We present some new results concerning perturbation theory for positive solutions of second-order linear elliptic operators, including further study of the equivalence of positive minimal Green functions and the validity of a Liouville…
We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in…
The aim of this paper is to consider the linear ultraparabolic equation with bounded and VMO coefficients $a_{ij} (z)$. Assume that the operator $L_0$ obtained by freezing the coefficients $a_{ij}(z)$ at any point ${z_0} \in {\mathbb{R}^{N…
The solvability in Sobolev spaces is proved for divergence form complex-valued higher order parabolic systems in the whole space, on a half space, and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable…
We prove weighted $L_{p,q}$-estimates for divergence type higher order elliptic and parabolic systems with irregular coefficients on Reifenberg flat domains. In particular, in the parabolic case the coefficients do not have any regularity…
In this paper, the aim of our work is to establish global weighted gradient estimates via fractional maximal functions and the point-wise regularity estimates of Dirichlet problem for divergence elliptic equations of the type \begin{align*}…
We introduce the concept of $C^{m,\alpha}$-nonlocal operators, extending the notion of second order elliptic operator in divergence form with $C^{m,\alpha}$-coefficients. We then derive the nonlocal analogue of the key existing results for…