Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-31T04:45:28.187Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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 first 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 the phenomenon of “crossing” of the exponents of singularities. For this case, we introduce the appropriate combinations of the simple tensor product singularities that allow us to give estimates in ordinary and weighted Sobolev spaces for the regular part of the solution and for the coefficients of the singularities. These combinations appear in a natural way as sections of an analytic bundle above the edge. Their behaviour is described with the help of divided differences of powers of the distance to the edge. The class of operators considered includes second-order elliptic operators with analytic complex-valued coefficients with mixed Dirichlet, Neumann or oblique derivative conditions. With our description of the singularities we are able to remove some restrictive hypotheses that were previously made in other works. In this first part, we prove the basic facts in a simplified framework. Nevertheless the tools we use are essentially the same in the general situation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 1 , 1993 , pp. 109 - 155
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.. Edge asymptotics on a skew cylinder. In Symposium ‘Analysis in Domains and on Manifolds with Singularities’, Breitenbrunn 1990, pp. 28–42, eds Schulze, W. and Triebel, H., Mathematik, Teubner-Texte zur, 131 (Leipzig: B. G. Teubner, 1992).Google Scholar
2Costabel, M. and Dauge, M.. Développement asymptotique le long d'une arête pour des équations elliptiques d'ordre 2 dans R3. C. R. Acad. Sci. Paris, Sér. I 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).CrossRefGoogle Scholar
5Kondrat'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
6Kondrat'ev, V. A.. Singularities of a solution of Dirichlet's problem for a second order elliptic equation in a neighborhood of an edge. Differential Equations 13 (1970), 1411–1415.Google Scholar
7Kondrat'ev, V. A. and Oleinik, O. A.. Boundary-value problems for partial differential equations in non-smooth domains. Russian Math. Surveys 38 (1983), 1–86.CrossRefGoogle Scholar
8Kufner, A. and Sändig, A. M.. Some applications of weighted Sobolev spaces (Leipzig: B. G. Teubner Verlagsgesellschaft, 1987).Google Scholar
9Maz'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
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 Babusška (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.CrossRefGoogle Scholar
12Nikishkin, A.. Singularities of the solution of the Dirichlet problem for a second order equation in a neighborhood of an edge. Moscow Univ. Math. Bull. 34(2) (1979), 53–64.Google Scholar
13Petersdorff, T. v. and Stephan, E. P.. Singularities of the solution of the Laplacian in domains with circular edges (to appear).Google Scholar
14Rempel, S.. Regularity of solutions to degenerate operators. Comm. Partial Differential Equations 14(7) (1989), 833–865.CrossRefGoogle Scholar
15Rempel, S. and Schulze, B. W.. Asymptotics for Elliptic Mixed Boundary Problems (Berlin: Akademie, 1989).CrossRefGoogle Scholar
16Schmutzler, B.. About the structure of branching asymptotics for elliptic boundary value problems in domains with edges. In Symposium ‘Analysis in Domains and on Manifolds with Singularities’, Breitenbrunn 1990, pp. 202–8, eds Schulze, W. and Triebel, H., Mathematik, Teubner-Texte zur, 131 (Leipzig: B. G. Teubner, 1992).Google Scholar
17Schulze, B. W.. Regularity with continuous and branching asymptotics for elliptic operators on manifold with edges. Integral Equations and Operator Theory 11(4) (1988), 557–602.CrossRefGoogle Scholar
18Triebel, H.. Interpolation theory. Function spaces. Differential operators, North-Holland Mathematical Library (Amsterdam: North-Holland, 1978).Google Scholar