### 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 W^{1} _{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(A^{αβ} _{(x1)Dβu)} = f , which typically arises in homogenization of layered materials. We relax the condition on f from H^{k}, k ≥ d/2, to Lp with p > d.

Original language | English |
---|---|

Pages (from-to) | 1075-1098 |

Number of pages | 24 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 43 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2011 Jul 21 |

Externally published | Yes |

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### 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.

Research output: Contribution to journal › Article

*SIAM Journal on Mathematical Analysis*, vol. 43, no. 3, pp. 1075-1098. https://doi.org/10.1137/100794614

}

TY - JOUR

T1 - Parabolic and elliptic systems in divergence form with variably partially bmo coefficients

AU - Dong, Hongjie

AU - Kim, Doyoon

PY - 2011/7/21

Y1 - 2011/7/21

N2 - 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.

AB - 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.

KW - Bounded mean oscillation

KW - Second order systems

KW - Sobolev spaces

KW - Variably partially BMO coefficients

UR - http://www.scopus.com/inward/record.url?scp=79960416777&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960416777&partnerID=8YFLogxK

U2 - 10.1137/100794614

DO - 10.1137/100794614

M3 - Article

AN - SCOPUS:79960416777

VL - 43

SP - 1075

EP - 1098

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 3

ER -