WEAK DIFFERENTIABILITY FOR SOLUTIONS TO NONLINEAR ELLIPTIC-SYSTEMS WITH P, Q-GROWTH CONDITIONS

We consider a vector-valued function u is-an-element-of W(loc)1, q(OMEGA; R(N)), OMEGA subset-of R(n), which is a weak solution of the elliptic system - SIGMA(i = 1)n D(i) {a(alpha)i(x, Du(x))} = 0, alpha = 1, ..., N. If the so-called <<p, q-growth conditions>> hold, then we prove that: (1 + \Du\2)P/4 is-an-element-of W(loc)1, 2(OMEGA); x --> a(alpha)i(x, Du(x)) is-an-element-of W(loc)1, q/(q - 1)(OMEGA); u is-an-element-of W(loc)2, 2(OMEGA; R(N)).