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 < ∞.