Geodesic
Anisotropic classes of uniqueness of the solution of the Dirichlet
Izvestiya. Mathematics , Tome 70 (2006) no. 6, pp. 1165-1200

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We select a class of uniqueness of the solutions of the quasi-elliptic equation with the Dirichlet condition on the boundary of an unbounded domain $\Omega\subset\mathbb R^{n+1}$ and show that for domains with irregular behaviour of the boundary this class can be wider than that established in [10] for second-order elliptic equations. For the Laplace equation we construct an example of non-uniqueness of solution of the Dirichlet problem that shows that the class of uniqueness found in this paper cannot be essentially extended. 
@article{IM2_2006_70_6_a3,
     author = {L. M. Kozhevnikova},
     title = {Anisotropic classes of uniqueness of the solution of the {Dirichlet}},
     journal = {Izvestiya. Mathematics },
     pages = {1165--1200},
     publisher = {mathdoc},
     volume = {70},
     number = {6},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2006_70_6_a3/}
}
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L. M. Kozhevnikova. Anisotropic classes of uniqueness of the solution of the Dirichlet. Izvestiya. Mathematics , Tome 70 (2006) no. 6, pp. 1165-1200. http://geodesic.mathdoc.fr/item/IM2_2006_70_6_a3/