A~class of degenerate elliptic operators

A. V. Fursikov

Sbornik. Mathematics, Tome 8 (1969) no. 3, pp. 357-382

Résumé

In a bounded region $G\subset R^n$ we consider an operator $A$ which is elliptic inside the region and degenerate on its boundary $\Gamma$. More precisely, the operator $A$ has the following form in the local coordinate system $(x',x_n)$, in which the boundary $\Gamma$ is given by the equation $x_n=0$ and $x_n>0$ for points in the region $G$: $$ Au=\sum_{|l'|+l_n+\beta\leqslant2m}a_{l',l_n,\beta}(x',x_n)q^\beta x_n^{l_n}D_{x'}^{l'}D_{x_n}^{l_n}u $$ where $q$ is a parameter, and $$ \sum_{|l'|+l_n+\beta=2m}a_{l',l_n,\beta}(x',0)q^\beta{\xi'}^{l'}{\xi_n}^{l^n}\ne0\quad\text{for}\quad|\xi|+|q|\ne0. $$ The operator $A$ will be proved Noetherian in certain spaces under the condition that $|q|$ is sufficiently large. In addition, some results will be obtained relating to how the smoothness of the solution of the equation $Au=f$ depends on the magnitude of the parameter. A theorem is formulated concerning unique solvability in approperiate spaces for a class of degenerate parabolic operators. Bibliography: 8 titles.