A priori estimates and smoothness of solutions of a system of quasi-linear equations that is elliptic in the Douglis-Nirenberg sense

We study a Douglis-Nirenberg elliptic system of quasi-linear equations. We solve the problem of the limiting admissible rate of growth of the non-linear terms of the system with respect to their arguments consistent with the possibility of obtaining estimates of the derivatives of a solution in terms of its maximum absolute value. The restrictions on the smoothness of the non linear terms are minimal and the results are sharp. We construct an example that shows the optimality of the upper bound for the exponent of growth. A priori L(p)-estimates are obtained both inside the domain and all the way to its boundary when the solution satisfies nonlinear boundary conditions of Lopatinskii type. We study the problem of smoothness inside the domain for solutions belonging to certain Sobolev spaces. We obtain estimates of the Holder norms of the derivatives of a solution. We prove a theorem on a removable isolated singularity of bounded solutions of general elliptic systems of quasi-linear equations. All results are new, even for a single second-order equation.