## Abstract

The strongly elliptic system (Formula presented.) with constant m × m matrix-valued coefficients (Formula presented.) for a vector-valued functions u = (u _{1} , …, u _{m} ) in the half-space (Formula presented.) as well as in a domain (Formula presented.) with smooth boundary ∂ Ω and compact closure (Formula presented.) is considered. A representation for the sharp constant (Formula presented.) in the inequality (Formula presented.) is obtained, where |·| is the length of a vector in the m-dimensional Euclidean space, (Formula presented.), and ‖·‖ _{p} is the L ^{p} -norm of the modulus of an m-component vector-valued function, 1 ≤p ≤∞. It is shown that (Formula presented.) where (Formula presented.) is a point at ∂ Ω nearest to x ∈ Ω, u is the solution of Dirichlet problem in Ω for the strongly elliptic system (Formula presented.) with boundary data from (Formula presented.), and (Formula presented.) is the sharp constant in the aforementioned inequality for u in the tangent space (Formula presented.) to ∂ Ω at (Formula presented.). As examples, Lamé and Stokes systems are considered. For instance, in the case of the Stokes system, the explicit formula (Formula presented.) is derived, where 1 < p < ∞.

Original language | English |
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Pages (from-to) | 783-805 |

Number of pages | 23 |

Journal | International Journal of Phytoremediation |

Volume | 86 |

Issue number | 7 |

DOIs | |

State | Published - Jul 2007 |

## Keywords

- 2000 Mathematics Subject Classifications: 35J55
- 35Q30
- 35Q72
- Boundary L -data
- Lamé and Stokes systems
- Pointwise estimates
- Strongly elliptic systems

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