Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-22T06:21:41.307Z Has data issue: false hasContentIssue false
  • English
  • Français

General edge asymptotics of solutions of second-order elliptic boundary value problems II

Published online by Cambridge University Press:  14 November 2011

Martin Costabel  and
Monique Dauge
Martin Costabel
Affiliation:
IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France
Monique Dauge
Affiliation:
Département de Mathématiques, Université de Nantes, 2, rue de la Houssinière, 44072 Nantes Cedex 03, France
Get access Rights & Permissions [Opens in a new window]

Synopsis

This is the second of two papers in which we study the singularities of solutions of second-order linear elliptic boundary value problems at the edges of piecewise analytic domains in ℝ3. When the opening angle at the edge is variable, there appears trie phenomenon of “crossing” of the exponents of singularities. In Part I, we introduced for the Dirichlet problem appropriate combinations of the simple tensor product singularities.

In this second part, we extend the results of Part I to general non-homogeneous boundary conditions. Moreover, we show how these combinations of singularities appear in a natural way as sections of an analytic vector bundle above the edge. In the case when the interior operator is the Laplacian, we give a simpler expression of the combined singular functions, involving divided differences of powers of a complex variable describing the coordinates in the normal plane to the edge.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 1 , 1993 , pp. 157 - 184
Copyright
Copyright © Royal Society of Edinburgh 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Costabel, M. and Dauge, M.. General edge asymptotics of solutions of second order elliptic boundary value problems I. Proc. Roy. Soc. Edinburgh, 000–000, 1992.Google Scholar
2Costabel, M. and Dauge, M.. Développement asymptotique le long d'une arête pour des équations elliptiques d'ordre 2 dans R 3. C. R. Acad. Sci. Paris, Sér. 1 Math. 312 (1991), 227–232.Google Scholar
3Dauge, M.. Problème de Dirichlet sur un polyèdre de R3 pour un opérateur fortement elliptique. Séminaire Equations aux Dérivées Partielles, Université de Nantes 1982–1983, No 5.Google Scholar
4Dauge, M.. Elliptic Boundary Value Problems in Corner Domains – Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics 1341 (Berlin: Springer, 1988).Google Scholar
5Dauge, M.. Higher order oblique derivative problems on polyhedral domains. Comm. Partial Differential Equations 8–9 (1989), 1193–1227.CrossRefGoogle Scholar
6Kondrat'ev, V. A.. Boundary-value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16 (1967), 227–313.Google Scholar
7Lehman, R. S.. Developments at an analytic corner of solutions of elliptic partial differential equations. J. Math. Mech. 8 (1959), 727–760.Google Scholar
8Maz'ya, V. G. and Plamenevskii, B. A.. Lp estimates of solutions of elliptic boundary value problems in a domain with edges. Trans. Moscow Math. Soc. 1 (1980), 49–97.Google Scholar
9Maz'ya, V. G. and Plamenevskii, B. A.. On boundary value problems for a second order elliptic equation in a domain with edges. Vestnik Leningrad Univ. Math. 8 (1980), 99–106.Google Scholar
10Maz'ya, V. G. and Rossmann, J.. Über die Asymptotik der Lösungen elliptischer Randwertaufgaben in der Umgebung von Kanten. Math. Nachr. 138 (1988), 27–53.CrossRefGoogle Scholar
11Maz'ya, V. G. and Rossmann, J.. On a problem of Babuška (Stable asymptotics of the solution to the Dirichlet problem for elliptic equations of second order in domains with angular points) Math. Nachr. 155 (1992), 199–220.Google Scholar
12Nazarov, S. A. and Plamenevskii, B. A.. Zadacha Neymana dlya samosopryazhennykh ellipticheskykh sistem v oblasti s kusochno-gladkoy granitsey (The Neumann problem for selfadjoint elliptic systems in domains with piecewise smooth boundaries, Russian). Trudy Leningr. mat. o-va. 1 (1991), 174–211.Google Scholar
13Stummel, F.. Rand- und Eigenwertaufgaben in Sobolevschen Räumen, Lecture Notes in Mathematics 102 (Berlin: Springer, 1969).Google Scholar