Cancelier, C. Problèmes aux limites pseudo-différentiels donnant lieu au principe du maximum. (Pseudodifferential boundary problems admitting a maximum principle). (French) Zbl 0646.35080 Commun. Partial Differ. Equations 11, 1677-1726 (1986). Let V be a \(C^{\infty}\) compact manifold and \(M\subset V\) be a compact \(C^{\infty}\) submanifold with boundary. Let P be a real second order degenerate elliptic differential operator of order 2 with \(C^{\infty}\) coefficients, and S be a real pseudo-differential operator of order \(2- \alpha\) \((\alpha >0)\) on V. Put \(S_ Mu=S\overset \circ u_{| M}\) if \({\overset \circ u}\) is the extension of \(u\) by 0 outside \(M\). The author studies existence, uniqueness and regularity up to the boundary of weak solutions of boundary problems of the type \[ Pu+S_ M u-\beta u=f \text{ in } M\quad (\beta >0); \quad Lu=0 \text{ in } \partial M, \] for which there is a maximum principle. Here \(L\) is either the usual trace operator or a Boutet de Monvel trace operator of order 2. Associated semi-groups are considered. Earlier results of Bony, Courrège and Priouret are generalized. Reviewer: P.Godin Cited in 1 ReviewCited in 9 Documents MSC: 35S15 Boundary value problems for PDEs with pseudodifferential operators 35B50 Maximum principles in context of PDEs 58J40 Pseudodifferential and Fourier integral operators on manifolds 35J25 Boundary value problems for second-order elliptic equations 35J70 Degenerate elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000) Keywords:compact manifold; existence; uniqueness; regularity; weak solutions; maximum principle; trace operator; semi-groups PDFBibTeX XMLCite \textit{C. Cancelier}, Commun. Partial Differ. Equations 11, 1677--1726 (1986; Zbl 0646.35080) Full Text: DOI References: [1] Bony J.–M., Ann. Inst. Fourier Grenoble 18 pp 369– (1968) · Zbl 0181.11704 [2] Boutet de Monvel L., Acta Math 126 pp 10– (1971) [3] Cancelier C., C.R. Acad. Sc. Paris, t 284 pp 795– (1977) [4] Cancelier C., C.R. Acad. Sc. Paris, t 288 pp 757– (2051) [5] C. Cancelier : Problème de Dirichlet intégro–différentiel et semi–groupe de Feller sur un ouvert borné de Rn, Conf. n{\(\deg\)}15, E.D.P. St-Jean-de-Monts. · Zbl 0445.45016 [6] Cancelier C., Problèmes aux limites pseudo–différentiels donnant lieu au principe du maximum (1984) [7] DOI: 10.1002/cpa.3160180305 · Zbl 0125.33302 [8] DOI: 10.1002/cpa.3160200410 · Zbl 0153.14503 [9] Oleinik O.A., Amer. Math. Soc. Trans 65 pp 167– (1967) · Zbl 0179.43102 [10] Oleinik O.A., Amer. Math. Soc (1973) [11] Rempel S., Index theory of elliptic boundary problems (1982) · Zbl 0504.35002 [12] Taira K., Problèmes aux limites non coercifs et applications (1978) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.