Geodesic
Entropy solutions of the Dirichlet problem for a~class of non-linear elliptic fourth-order equations with right-hand sides in~$L^1$
Izvestiya. Mathematics , Tome 65 (2001) no. 2, pp. 231-283

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In this paper we introduce and study the notion of an entropy solution of the Dirichlet problem for a class of non-linear elliptic fourth-order equations whose right-hand sides admit arbitrary growth with respect to the variable corresponding to the unknown function and belong to the space $L^1$ for each fixed value of this variable. We prove the existence and uniqueness of an entropy solution. We establish the existence of so-called $H$-solutions and $W$-solutions of the problem and prove that the entropy solutions belong to certain Sobolev spaces. 
@article{IM2_2001_65_2_a1,
     author = {A. A. Kovalevsky},
     title = {Entropy solutions of the {Dirichlet} problem for a~class of non-linear elliptic fourth-order equations with right-hand sides in~$L^1$},
     journal = {Izvestiya. Mathematics },
     pages = {231--283},
     publisher = {mathdoc},
     volume = {65},
     number = {2},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2001_65_2_a1/}
}
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A. A. Kovalevsky. Entropy solutions of the Dirichlet problem for a~class of non-linear elliptic fourth-order equations with right-hand sides in~$L^1$. Izvestiya. Mathematics , Tome 65 (2001) no. 2, pp. 231-283. http://geodesic.mathdoc.fr/item/IM2_2001_65_2_a1/