Hypoellipticity for a class of degenerate elliptic operators of second order
type: Item
instance of: scholarly article
title: Hypoellipticity for a class of degenerate elliptic operators of second order
zbMATH Open document ID: 0804.35026
author: Michiharu Suzuki
published in: Tsukuba Journal of Mathematics
publication date: 1993-02-22T00:00:00Z
review text: The paper under consideration is devoted mainly to the investigation of some microhypoelliptic properties for the following class of degenerate elliptic partial differential operators of second order \(p(x,D) = D^ 2_ 1 + \alpha (x)D^ 2_ 2 + \beta (x) D_ 2\), \(x \in \mathbb{R}^ 2\), \(\alpha (x) \geq 0\), \(\alpha (0) = 0\). Under several assumptions for subordination of \(\text{Re} \beta\), \(\text{Im} \beta\) to \(\alpha\), respectively for the geometrical structure of the set \(S = \{x | \alpha (x) = 0\}\) microhypoellipticity is proved. To illustrate the results here obtained the next example will be proposed. Example. Assume \(S = \{x | f(x) = 0\}\), where \(df(0) \neq 0\), \((\partial f/ \partial x_ 1)(0) = 0\), \((\partial f/ \partial x_ 1)(x) \neq 0\) if \(x \neq 0\). Moreover, let \((\text{Re} \beta)^ 2 + (\text{Im} \beta)^{2\ell} \leq C \alpha\) for some \(C>0\) and \(\ell \in \mathbb{N}\). Then \(P\) is microhypoelliptic in \(\mathbb{R}^ 2\). The main theorems of the author are a further development and generalization of some papers of Hörmander, Fedii, Morimoto, Kajitani and Wakabayashi.
Mathematics Subject Classification ID: 35H10
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35S05
zbMATH DE Number: 109340
zbMATH Keywords: degenerate elliptic operators of second order
zbMATH Keywords: microhypoellipticity
reviewed by: Peter R. Popivanov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496161842
OpenAlex ID: W2265814623
DOI: 10.21099/TKBJM/1496161842
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description: scientific article; zbMATH DE number 109340