On~a~class of globally hypoelliptic operators

A. V. Fursikov

Sbornik. Mathematics, Tome 20 (1973) no. 3, pp. 383-405

Résumé

We consider an operator $A$ which is defined on an $(n+1)$-dimensional manifold $\Omega$ and which is elliptic everywhere outside an $n$-dimensional submanifold $\Gamma$. If $(x)$ represents the local coordinates in $\Gamma$ and $t$ is the distance to $\Gamma$, then in the coordinates $(x,t)$ the operator $A$ is of the form $$ Au=\sum_{|\beta|+l\leqslant m}a_{\beta l}(x,t)t^{lq}D^\beta_xD^l_tu, $$ where $q>1$ is an integer. We present a necessary and sufficient condition for infinite differentiability in a neighborhood of $\Gamma$ of the solution of $Au=f$ if $f$ is infinitely differentiable in a neighborhood of $\Gamma$. Bibliography: 16 titles.