The strongly elliptic system Aij partial derivative(2)u/partial derivative x(i)partial derivative x(j) = 0 with constant m x m matrix-valued coefficients A(ij) = A(ji) for a vector-valued functions u = (u(1),...,u(m)) in the half-space R-+(n) = {x = (x(1),..., x(n)) : x(n) > 0} as well as in a domain Omega subset of R-n with smooth boundary partial derivative Omega and compact closure Omega is considered. A representation for the sharp constant C-p in the inequality vertical bar u(x)vertical bar <= C(p)x(n)((1-n)/p) parallel to u vertical bar x(n)=0 parallel to p is obtained, where vertical bar center dot vertical bar is the length of a vector in the m-dimensional Euclidean space, x epsilon R-+(n), and parallel to center dot parallel to(p) is the L-p-norm of the modulus of an m-component vector-valued function, 1 <= p <= infinity. It is shown that lim vertical bar x - O-x vertical bar((n-1)/p) sup{vertical bar u(x)vertical bar : parallel to u vertical bar partial derivative Omega parallel to p <= 1} =C-p(O-x), x -> O-x where O-x is a point at partial derivative Omega nearest to x epsilon Omega, u is the solution of Dirichlet problem in Omega for the strongly elliptic system A(ij)partial derivative(2)u/partial derivative x(i)partial derivative x(j) = 0 with boundary data from [L-p(partial derivative Omega)](m), and C-p(O-x) is the sharp constant in the aforementioned inequality for u in the tangent space R-+(n) (O-x) to partial derivative Omega at O-x. As examples, Lame ' and Stokes systems are considered. For instance, in the case of the Stokes system, the explicit formula C-p = 2 Gamma((n + 2)/2)/pi((n+p-1)/(2p)) {Gamma((2p + n - 1)/(2p - 2))/Gamma((n + 1)p/(2p - 2))}((p-1)/p) is derived, where 1 < p < infinity.