New a Priori Estimates for q-Nonlinear Elliptic Systems with Strong Nonlinearities in the Gradient, 1<q<2

We consider q-nonlinear nondiagonal elliptic systems, where 1 < q < 2, with strong nonlinear terms in the gradient. Under a smallness condition on the gradient of a solution in the Morrey space L^q,n−q, we estimate the L^p-norm of the gradient for p > q and the Holder norm of the solution for the case n = 2. An abstract theorem on “quasireverse Holder inequalities” proved by the author earlier is used essentially. Bibliography: 24 titles. Dedicated to N. N. Uraltseva on the occasion of her 70th birthday __________ Published in Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 19–48.