Parabolic and elliptic systems in divergence form with variably partially bmo coefficients

Hongjie Dong, Doyoon Kim

Research output: Contribution to journalArticle

32 Citations (Scopus)

Abstract

We establish the solvability of second order divergence-type parabolic systems in Sobolev spaces. The leading coefficients are assumed to be merely measurable in one spatial direction on each small parabolic cylinder with the spatial direction allowed to depend on the cylinder. In the other orthogonal directions and the time variable, the coefficients have locally small mean oscillations. We also obtain the corresponding W1 p -solvability of second order elliptic systems in divergence form. This type of system arises from the problems of linearly elastic laminates and composite materials. Our results are new even for scalar equations, and the proofs differ from and simplify the methods used previously in [H. Dong and D. Kim, Arch. Ration. Mech. Anal., 196 (2010), pp. 25-70]. As an application, we improve a result by Chipot, Kinderlehrer, and Vergara-Caffarelli [Arch. Ration. Mech. Anal., 96 (1986), pp. 81-96] on gradient estimates for elasticity system Da(Aαβ (x1)Dβu) = f , which typically arises in homogenization of layered materials. We relax the condition on f from Hk, k ≥ d/2, to Lp with p > d.

Original languageEnglish
Pages (from-to)1075-1098
Number of pages24
JournalSIAM Journal on Mathematical Analysis
Volume43
Issue number3
DOIs
Publication statusPublished - 2011 Jul 21
Externally publishedYes

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Parabolic Systems
Arches
Elliptic Systems
Divergence
Arch
Sobolev spaces
Solvability
Coefficient
Laminates
Elasticity
Gradient Estimate
Second-order Systems
Composite Materials
Type Systems
Homogenization
Sobolev Spaces
Simplify
Composite materials
Linearly
Scalar

Keywords

  • Bounded mean oscillation
  • Second order systems
  • Sobolev spaces
  • Variably partially BMO coefficients

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Computational Mathematics

Cite this

Parabolic and elliptic systems in divergence form with variably partially bmo coefficients. / Dong, Hongjie; Kim, Doyoon.

In: SIAM Journal on Mathematical Analysis, Vol. 43, No. 3, 21.07.2011, p. 1075-1098.

Research output: Contribution to journalArticle

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