Sharp pointwise estimates for solutions of strongly elliptic second order systems with boundary data from L^p

Abstract

The strongly elliptic system with constant m × m matrix-valued coefficients for a vector-valued functions u  = (u ₁, … , u _m ) in the half-space as well as in a domain with smooth boundary ∂ Ω and compact closure is considered. A representation for the sharp constant in the inequality

is obtained, where |·| is the length of a vector in the m-dimensional Euclidean space, , and ‖·‖ _p is the L ^p -norm of the modulus of an m-component vector-valued function, 1 ≤p ≤∞.

 It is shown that

where is a point at ∂ Ω nearest to x ∈ Ω, u is the solution of Dirichlet problem in Ω for the strongly elliptic system with boundary data from , and is the sharp constant in the aforementioned inequality for u in the tangent space to ∂ Ω at . As examples, Lamé and Stokes systems are considered. For instance, in the case of the Stokes system, the explicit formula

is derived, where 1 < p < ∞.

Keywords:

• Boundary L ^p -data
• Pointwise estimates
• Strongly elliptic systems
• Lamé and Stokes systems

Keywords:

• 2000 Mathematics Subject Classifications: 35J55
• 35Q30
• 35Q72

Acknowledgments

The research of G. Kresin was supported by the KAMEA program of the Ministry of Absorption, State of Israel, and by the College of Judea and Samaria, Ariel. V. Maz'ya was partially supported by NSF grant DMS 0500029.