Geodesic
A regularity condition for generalized solutions of higher-order quasilinear elliptic equations
Izvestiya. Mathematics , Tome 7 (1973) no. 6, pp. 1371-1421

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Regularity is proved for an arbitrary generalized solution of a quasilinear elliptic equation of divergent type which belongs to $W_2^{m+n/2}(\Omega')$, for an arbitrary strictly interior subregion $\Omega'$ of a region $\Omega$ ($2m$ is the order of the equation, and $n$ is the number of arguments). It follows from this, in particular, that the regularity problem has an affirmative solution in the two-dimensional case. 
@article{IM2_1973_7_6_a7,
     author = {I. V. Skrypnik},
     title = {A regularity condition for generalized solutions of higher-order quasilinear elliptic equations},
     journal = {Izvestiya. Mathematics },
     pages = {1371--1421},
     publisher = {mathdoc},
     volume = {7},
     number = {6},
     year = {1973},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1973_7_6_a7/}
}
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I. V. Skrypnik. A regularity condition for generalized solutions of higher-order quasilinear elliptic equations. Izvestiya. Mathematics , Tome 7 (1973) no. 6, pp. 1371-1421. http://geodesic.mathdoc.fr/item/IM2_1973_7_6_a7/