Geodesic
On a~degenerating problem with directional derivative
Sbornik. Mathematics, Tome 16 (1972) no. 3, pp. 429-469

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We study a problem with directional derivative for a second order elliptic equation. We assume that smooth compact submanifolds $\Gamma_0\supset\Gamma_1\supset\cdots\supset\Gamma_s$ have been selected from the boundary $\Gamma$, and that the vector field is tangent to $\Gamma_i$ ($i\leqslant s-1$) at points of $\Gamma_{i+1}$ and not tangent to $\Gamma_s$. We show that the problem has a unique solution, obtain estimates of the solutions in $L_p(\Gamma)$ ($1$), and prove that the inverse operator is compact. Bibliography: 29 titles. 
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     author = {V. G. Maz'ya},
     title = {On a~degenerating problem with directional derivative},
     journal = {Sbornik. Mathematics},
     pages = {429--469},
     publisher = {mathdoc},
     volume = {16},
     number = {3},
     year = {1972},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1972_16_3_a7/}
}
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V. G. Maz'ya. On a~degenerating problem with directional derivative. Sbornik. Mathematics, Tome 16 (1972) no. 3, pp. 429-469. http://geodesic.mathdoc.fr/item/SM_1972_16_3_a7/