Comptes Rendus
Partial differential equations/Geometry
Analysis and boundary value problems on singular domains: An approach via bounded geometry
Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 487-493.

We prove well-posedness and regularity results for elliptic boundary value problems on certain singular domains that are conformally equivalent to manifolds with boundary and bounded geometry. Our assumptions are satisfied by the domains with a smooth set of singular cuspidal points, and hence our results apply to the class of domains with isolated oscillating conical singularities. In particular, our results generalize the classical L2-well-posedness result of Kondratiev for the Laplacian on domains with conical points. However, our domains and coefficients are too general to allow for singular function expansions of the solutions similar to the ones in Kondratiev's theory. The proofs are based on conformal changes of metric, on the differential geometry of manifolds with boundary and bounded geometry, and on our earlier geometric and analytic results on such manifolds.

Nous montrons des résultats de solvabilité et de régularité pour des problèmes de valeurs au bord elliptiques dans certains domaines singuliers, conformément équivalents aux variétés avec bord et géométrie bornée. Nos hypothèses sont satisfaites par les domaines ayant un ensemble lisse de points cuspidaux singuliers, et nos résultats s'appliquent donc à la classe des domaines à singularités coniques oscillantes isolées. En particulier, ils généralisent le résultat classique de Kondratiev de solvabilité L2 pour le laplacien sur les domaines à points coniques. Toutefois, nos domaines et coefficients sont trop généraux pour permettre de développer les solutions en fonctions singulières, comme avec la théorie de Kondratiev. Les démonstrations reposent sur des changements de métriques conformes, sur la géométrie des variétés à bord et géométrie bornée, ainsi que sur nos travaux antérieurs d'analyse et géométrie sur de telles variétés.

Published online:
DOI: 10.1016/j.crma.2019.04.009

Bernd Ammann 1; Nadine Große 2; Victor Nistor 3, 4

1 Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
2 Mathematisches Institut, Universität Freiburg, 79104 Freiburg, Germany
3 Université de Lorraine, CNRS, IECL, 57000 Metz, France
4 Institute of Mathematics of the Romanian Academy, PO BOX 1-764, 014700 Bucharest, Romania
     author = {Bernd Ammann and Nadine Gro{\ss}e and Victor Nistor},
     title = {Analysis and boundary value problems on singular domains: {An} approach via bounded geometry},
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Bernd Ammann; Nadine Große; Victor Nistor. Analysis and boundary value problems on singular domains: An approach via bounded geometry. Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 487-493. doi : 10.1016/j.crma.2019.04.009.

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Cited by Sources:

B.A. and N.G. have been partially supported by SPP 2026 (Geometry at infinity), funded by the DFG. B.A. has also been partially supported by the DFG SFB 1085 (Higher Invariants). V.N. has been partially supported by ANR 14-CE25-0012-01 (SINGSTAR).

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