Solvability problem for nondiagonal elliptic systems with a quadratic nonlinearity in the gradient (The two-dimensional case)

The existence of a weak solution of the Dirichlet problem for nondiagonal elliptic systems with quadratic nonlinearity is proved in the two-dimensional case. The solution is established to be smooth in the closure of a given domain, with the exception of at most finitely many points. The result is essentially based upon the theorem on quasireverse Hölder inequalities previously proved by the author. Bibliography: 14 titles.Dedicated to Academician O. A. Ladyzhenskaya on the occasion of her jubilee__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 5–17.