A priori estimates near the boundary for the solutions of non-diagonal elliptic systems with strong non-linearity

We consider quasilinear elliptic non-diagonal systems of equations with strong non-linearity with respect to the gradient. We have already shown that the generalized solution of this problem is Hölder continuous in the neighbourhood of points of the domain at which the norm of the gradient of the solution is sufficiently small in the Morrey space $L^{2,n-2}$. We estimate the Hölder norm of the solution in the neighbourhood of such points in terms of its norm in the Sobolev space $W_2^1$. We obtain a similar result under the Dirichlet boundary condition for points situated in the neighbourhood of the boundary.