On a degenerating problem with directional derivative

V. G. Maz'ya

Sbornik. Mathematics, Tome 16 (1972) no. 3, pp. 429-469

Résumé

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 $L_p(\Gamma)$ ($1<p\leqslant\infty$), and prove that the inverse operator is compact. Bibliography: 29 titles.

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