Geodesic
Solvability of nondiagonal elliptic systems with quadratic growth nonlinearities (two-dimensional case)
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 33, Tome 295 (2003), pp. 5-17

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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. 
@article{ZNSL_2003_295_a0,
     author = {A. A. Arkhipova},
     title = {Solvability of nondiagonal elliptic systems with quadratic growth nonlinearities (two-dimensional case)},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--17},
     publisher = {mathdoc},
     volume = {295},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a0/}
}
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A. A. Arkhipova. Solvability of nondiagonal elliptic systems with quadratic growth nonlinearities (two-dimensional case). Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 33, Tome 295 (2003), pp. 5-17. http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a0/