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

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 < ∞.