Estimates For Elliptic Systems Research Articles

$ W^{1,p} $ estimates for elliptic systems on composite material with almost partially BMO coefficients

<p style='text-indent:20px;'>In this paper, we establish uniform <inline-formula><tex-math id="M1">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates for composite material problems which can be described by a divergence form elliptic system on a nonsmooth domain composed of a finite number of subdomains. We want to derive global <inline-formula><tex-math id="M2">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> regularity under the assumption that the coefficients are almost <inline-formula><tex-math id="M3">\begin{document}$ (\delta,R) $\end{document}</tex-math></inline-formula>-vanishing of codimension 1 (see Definition 1.2) in each of multiple subdomains and the boundaries of subdomains are Reifenberg flat, moreover the estimates do not depend on the distance between these subdomains.</p>

A note on estimates for elliptic systems with L1 data

In this paper, we give necessary and sufficient conditions on the compatibility of a kth-order homogeneous linear elliptic differential operator A and differential constraint C for solutions toAu=fsubject toCf=0 in Rnto satisfy the estimates‖Dk−ju‖Lnn−j(Rn)⩽c‖f‖L1(Rn)for j∈{1,…,min⁡{k,n−1}} and‖Dk−nu‖L∞(Rn)⩽c‖f‖L1(Rn)when k≥n.

Estimates for elliptic systems in a narrow region arising from composite materials

In this paper, we establish the pointwise upper and lower bounds of the gradients of solutions to a class of elliptic systems, including linear systems of elasticity, in a general narrow region and in all dimensions. This problem arises from the study of damage analysis of high-contrast composite materials. Our results show that the damage may initiate from the narrowest place.

Gradient estimates for elliptic systems with measurable coefficients in nonsmooth domains

We consider an elliptic system in divergence form with measurable coefficients in a nonsmooth bounded domain to find a minimal regularity requirement on the coefficients and a lower level of geometric assumption on the boundary of the domain for a global W1,p, 1 < p < ∞, regularity. It is proved that such a W1,p regularity is still available under the assumption that the coefficients are merely measurable in one variable and have small BMO semi-norms in the other variables while the domain can be locally approximated by a hyperplane, a so called δ-Reifenberg domain, which is beyond the Lipschitz category. This regularity easily extends to a certain Orlicz-Sobolev space.

Estimates for elliptic systems from composite material

Estimates for elliptic systems from composite material

Interior Regularity Estimates for Elliptic Systems of Difference Equations

We prove interior regularity estimates for a large class of difference approximations for elliptic systems of partial differential equations. These estimates are analogous to those for the systems of differential equations. From these regularity estimates we obtain estimates on the convergence of the solutions of the difference equations. A theory of pseudo-difference operators with order is used to prove the regularity results. We also comment on factors to be considered in choosing a difference scheme for computations.