Solvability of nondiagonal elliptic systems with quadratic growth nonlinearities (two-dimensional case)

Existence of a weak solution of the Dirichlet problem to nondiagonal elliptic systems with quadratic growth nonlinearities is proved in the two-dimensional case. It is established that the solution is smooth in the closure of a given domain with exception of at most finitely many points. The result is essentially based upon the theorem on “quasireverse” Hölder inequalities earlier proved by the author.