Boundary value problems for elliptic equations degenerate on the boundary of a~domain

M. I. Vishik ; V. V. Grushin

Sbornik. Mathematics, Tome 9 (1969) no. 4, pp. 423-454

Résumé

We investigate the elliptic equation $Lu=f$ of order $2m$, degenerate on the boundary $\Gamma$ of a bounded domain $G$. In local coordinates $(x_1,\dots,x_n)$, introduced in a neighborhood $U(x_0)$ of the point $x_0\in\Gamma$ and in which $\Gamma\cap U(x_0)$ is given by $x_n=0$, the operator $$ L(x;x_n;D^\alpha)=\sum_{|\alpha|\leqslant m}\alpha_\alpha(x)x_n^{l_\alpha}D^\alpha, $$ where $l_\alpha=\max(0,q\alpha_n+q'\alpha'-qr)$, $q\geqslant1$, $q'\geqslant0$. For $x_n=0$ the operator $Lu$ degenerates into the quasi-elliptic operator $$ L_1u=\sum_{\frac rr'|\alpha'|+\alpha_n\leqslant r}\alpha_\alpha(x)D^\alpha\qquad(|\alpha'|\leqslant r'\quad(qr=q'r')). $$ In particular we study the case of transition, for $x_n=0$, of an elliptic operator into a parabolic operator. Figures: 3. Bibliography: 19 titles.