On the solvability of certain degenerate partial differential operators. (English) Zbl 1378.35004
Colombini, Ferruccio (ed.) et al., Shocks, singularities and oscillations in nonlinear optics and fluid mechanics. Papers based on the workshop, Rome, Italy, September 2015. Cham: Springer (ISBN 978-3-319-52041-4/hbk; 978-3-319-52042-1/ebook). Springer INdAM Series 17, 151-179 (2017).
The paper is a survey on recent results concerning local solvability, hypoellipticity and propagation of singularities for pseudo-differential operators with multiple characteristics. In particular, after recalling classical contributions on local solvability and propagation for operators of principal type, the author presents first the results from C. Parenti and the author [Int. Math. Res. Not. 2014, No. 14, 3790–3817 (2014; Zbl 1302.35493)], adding several examples and comments. Here transversally elliptic operators with symplectic multiple characteristics are considered, of the type of those by L. Boutet de Monvel et al. [Astérisque 34–35, 93–121 (1976; Zbl 0344.32009)].
In the case when the condition for minimal loss of derivatives is violated, precise results of solvability and propagation are obtained under suitable assumptions on the lower order terms.
The second part of the paper is devoted to the results from S. Federico and the author [Commun. Partial Differ. Equations 41, No. 3, 484–514 (2016; Zbl 1348.35055)], concerning the case when the characteristic manifold is obtained from the interaction of two kinds of degeneracies.
For the entire collection see [Zbl 1371.76002].
In the case when the condition for minimal loss of derivatives is violated, precise results of solvability and propagation are obtained under suitable assumptions on the lower order terms.
The second part of the paper is devoted to the results from S. Federico and the author [Commun. Partial Differ. Equations 41, No. 3, 484–514 (2016; Zbl 1348.35055)], concerning the case when the characteristic manifold is obtained from the interaction of two kinds of degeneracies.
For the entire collection see [Zbl 1371.76002].
Reviewer: Luigi Rodino (Torino)
MSC:
| 35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 35H10 | Hypoelliptic equations |
| 35A21 | Singularity in context of PDEs |
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