### Abstract

Let A_{1}(x, D) and A_{2}(x, D) be differential operators of the first order acting on l-vector functions u = (u^{1}, . . . , u^{1}) in a bounded domain Ω ⊂ ℝ^{n} with the smooth boundary ∂Ω. We assume that the H^{1}-norm, is equivalent to, where B_{i} = B_{i}(x, ν) is the trace operator onto ∂ Ω associated with A_{i}(x, D) for i = 1, 2 which is determined by the Stokes integral formula (ν: unit outer normal to ∂Ω. Furthermore, we impose on A_{1} and A_{2} a cancellation property such as A_{1}A_{2}′ = 0 and A_{2}A_{1}′ = 0, where A_{i}′ is the formal adjoint differential operator of A_{i}(i = 1, 2). Suppose that and converge to u and v weakly in L^{2}(Ω), respectively. Assume also that and are bounded in L^{2}(Ω). If either or is bounded in H^{1/2}(∂Ω), then it holds that. We also discuss a corresponding result on compact Riemannian manifolds with boundary.

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

Pages (from-to) | 879-905 |

Number of pages | 27 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 207 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2013 |

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### ASJC Scopus subject areas

- Analysis
- Mechanical Engineering
- Mathematics (miscellaneous)

### Cite this

*Archive for Rational Mechanics and Analysis*,

*207*(3), 879-905. https://doi.org/10.1007/s00205-012-0583-7

**Global Compensated Compactness Theorem for General Differential Operators of First Order.** / Kozono, Hideo; Yanagisawa, Taku.

Research output: Contribution to journal › Article

*Archive for Rational Mechanics and Analysis*, vol. 207, no. 3, pp. 879-905. https://doi.org/10.1007/s00205-012-0583-7

}

TY - JOUR

T1 - Global Compensated Compactness Theorem for General Differential Operators of First Order

AU - Kozono, Hideo

AU - Yanagisawa, Taku

PY - 2013

Y1 - 2013

N2 - Let A1(x, D) and A2(x, D) be differential operators of the first order acting on l-vector functions u = (u1, . . . , u1) in a bounded domain Ω ⊂ ℝn with the smooth boundary ∂Ω. We assume that the H1-norm, is equivalent to, where Bi = Bi(x, ν) is the trace operator onto ∂ Ω associated with Ai(x, D) for i = 1, 2 which is determined by the Stokes integral formula (ν: unit outer normal to ∂Ω. Furthermore, we impose on A1 and A2 a cancellation property such as A1A2′ = 0 and A2A1′ = 0, where Ai′ is the formal adjoint differential operator of Ai(i = 1, 2). Suppose that and converge to u and v weakly in L2(Ω), respectively. Assume also that and are bounded in L2(Ω). If either or is bounded in H1/2(∂Ω), then it holds that. We also discuss a corresponding result on compact Riemannian manifolds with boundary.

AB - Let A1(x, D) and A2(x, D) be differential operators of the first order acting on l-vector functions u = (u1, . . . , u1) in a bounded domain Ω ⊂ ℝn with the smooth boundary ∂Ω. We assume that the H1-norm, is equivalent to, where Bi = Bi(x, ν) is the trace operator onto ∂ Ω associated with Ai(x, D) for i = 1, 2 which is determined by the Stokes integral formula (ν: unit outer normal to ∂Ω. Furthermore, we impose on A1 and A2 a cancellation property such as A1A2′ = 0 and A2A1′ = 0, where Ai′ is the formal adjoint differential operator of Ai(i = 1, 2). Suppose that and converge to u and v weakly in L2(Ω), respectively. Assume also that and are bounded in L2(Ω). If either or is bounded in H1/2(∂Ω), then it holds that. We also discuss a corresponding result on compact Riemannian manifolds with boundary.

UR - http://www.scopus.com/inward/record.url?scp=84872945790&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84872945790&partnerID=8YFLogxK

U2 - 10.1007/s00205-012-0583-7

DO - 10.1007/s00205-012-0583-7

M3 - Article

AN - SCOPUS:84872945790

VL - 207

SP - 879

EP - 905

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 3

ER -