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},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {487--493},
     publisher = {Elsevier},
     volume = {357},
     number = {6},
     year = {2019},
     doi = {10.1016/j.crma.2019.04.009},
     language = {en},
AU  - Bernd Ammann
AU  - Nadine Große
AU  - Victor Nistor
TI  - Analysis and boundary value problems on singular domains: An approach via bounded geometry
JO  - Comptes Rendus. Mathématique
PY  - 2019
SP  - 487
EP  - 493
VL  - 357
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crma.2019.04.009
LA  - en
ID  - CRMATH_2019__357_6_487_0
ER  - 
%0 Journal Article
%A Bernd Ammann
%A Nadine Große
%A Victor Nistor
%T Analysis and boundary value problems on singular domains: An approach via bounded geometry
%J Comptes Rendus. Mathématique
%D 2019
%P 487-493
%V 357
%N 6
%I Elsevier
%R 10.1016/j.crma.2019.04.009
%G en
%F CRMATH_2019__357_6_487_0
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.

[1] H. Amann Function spaces on singular manifolds, Math. Nachr., Volume 286 (2013) no. 5–6, pp. 436-475

[2] H. Amann Uniformly regular and singular Riemannian manifolds, Elliptic and Parabolic Equations, Springer Proceedings in Mathematics & Statistics, vol. 119, Springer, Cham, Switzerland, 2015, pp. 1-43

[3] E. Amar Sobolev embeddings with weights in complete Riemannian manifolds (preprint) | arXiv

[4] B. Ammann; A.D. Ionescu; V. Nistor Sobolev spaces on Lie manifolds and regularity for polyhedral domains, Doc. Math., Volume 11 (2006), pp. 161-206 (electronic)

[5] B. Ammann; N. Große; V. Nistor Well-posedness of the Laplacian on manifolds with boundary and bounded geometry, Math. Nachr. (2019) (online first) | DOI

[6] B. Ammann; N. Große; V. Nistor The strong Legendre condition and the well-posedness of mixed Robin problems on manifolds with bounded geometry (Accepted in: Revue Roumaine de Mathématiques Pures et Appliquées, 2018, special issue “Spectral theory and applications to mathematical physics”) | arXiv

[7] H. Brezis Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011

[8] C. Băcuţă; A. Mazzucato; V. Nistor; L. Zikatanov Interface and mixed boundary value problems on n-dimensional polyhedral domains, Doc. Math., Volume 15 (2010), pp. 687-745

[9] G. Cardone; S.A. Nazarov; J. Sokolowski Asymptotics of solutions of the Neumann problem in a domain with closely posed components of the boundary, Asymptot. Anal., Volume 62 (2009) no. 1–2, pp. 41-88

[10] M. Costabel; M. Dauge; S. Nicaise Analytic regularity for linear elliptic systems in polygons and polyhedra, Math. Models Methods Appl. Sci., Volume 22 (2012) no. 8

[11] M. Dauge Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988

[12] M. Dauge Strongly elliptic problems near cuspidal points and edges, Partial Differential Equations and Functional Analysis, Progr. Nonlinear Differential Equations Appl., vol. 22, Birkhäuser Boston, Boston, MA, USA, 1996, pp. 93-110

[13] M. Disconzi; Y. Shao; G. Simonett Some remarks on uniformly regular Riemannian manifolds, Math. Nachr., Volume 289 (2016) no. 2–3, pp. 232-242

[14] N. Große; V. Nistor Uniform Shapiro–Lopatinski conditions and boundary value problems on manifolds with bounded geometry, Potential Anal. (2019) (online first) | DOI

[15] N. Große; C. Schneider Sobolev spaces on Riemannian manifolds with bounded geometry: general coordinates and traces, Math. Nachr., Volume 286 (2013) no. 16, pp. 1586-1613

[16] I.V. Kamotski; V.G. Maz'ya On the third boundary value problem in domains in cusps, J. Math. Sci. (N.Y.), Volume 173 (2011) no. 5, pp. 609-631

[17] V.A. Kondrat'ev Boundary value problems for elliptic equations in domains with conical or angular points, Transl. Moscow Math. Soc., Volume 16 (1967), pp. 227-313

[18] V.A. Kozlov; V.G. Maz'ya; J. Rossmann Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs, vol. 52, American Mathematical Society, Providence, RI, USA, 1997

[19] M. Lesch; N. Peyerimhoff On index formulas for manifolds with metric horns, Commun. Partial Differ. Equ., Volume 23 (1998) no. 3–4, pp. 649-684

[20] J.-L. Lions; E. Magenes Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, vol. 181, Springer-Verlag, New York–Heidelberg, 1972 (translated from the French by P. Kenneth)

[21] A. Munnier; K. Ramdani Asymptotic analysis of a Neumann problem in a domain with cusp. Application to the collision problem of rigid bodies in a perfect fluid, SIAM J. Math. Anal., Volume 47 (2015) no. 6, pp. 4360-4403

[22] S. Nazarov; N. Popoff Self-adjoint and skew-symmetric extensions of the Laplacian with singular Robin boundary condition, C. R. Acad. Sci. Paris, Ser. I, Volume 356 (2018) no. 9, pp. 927-932

[23] S.A. Nazarov; B.A. Plamenevsky Elliptic Problems in Domains with Piecewise Smooth Boundaries, de Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994

[24] V. Rabinovich; B.-W. Schulze; N. Tarkhanov C-algebras of singular integral operators in domains with oscillating conical singularities, Manuscr. Math., Volume 108 (2002) no. 1, pp. 69-90

[25] B.-W. Schulze; B. Sternin; V. Shatalov An operator algebra on manifolds with cusp-type singularities, Ann. Glob. Anal. Geom., Volume 16 (1998) no. 2, pp. 101-140

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).

Comments - Policy