The global hypoellipticity of degenerate elliptic-parabolic operators (Q583481)

Only few papers dealt with global hypoellipticity, for example, \textit{O. A. Oleinik} and \textit{E. V. Radkevich} [Second order equations with nonnegative characteristic form, Am. Math. Soc. (1973); see the review of the Russian original (1971; Zbl 0217.415)]. They proved the global hypoellipticity of the degenerate elliptic-parabolic operator \[ P=\sum^{d}_{i,j=1}a^{ij}\partial x_ i\partial x_ j+\sum^{d}_{i=1}b^ i\partial x_ j+c(x) \] (x\(\in \Omega \subset {\mathbb{R}}^ d\), \((a^{ij})_{i,j=1,...,d}\) positive definite), when \(S=\{x:\) dim Lie(x)\(<d\}\) is either a smooth hypersurface or an isolated point. Extending their results, this paper obtains sufficient conditions for global hypoellipticity stated in terms of diffusion vector field and drift vector field. The obtained theorems apply to the case when S is not a smooth hypersurface.