General edge asymptotics of solutions of second-order elliptic boundary value problems. I. (English) Zbl 0791.35032

Authors’ abstract: This is the first of two papers (see below) in which we study the singularities of solutions of second-order linear elliptic boundary value problems at the edges of piecewise analytic domains in \(\mathbb{R}^ 3\). 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.
Reviewer: H.Ding (Beijing)


35J25 Boundary value problems for second-order elliptic equations
35A20 Analyticity in context of PDEs
35D10 Regularity of generalized solutions of PDE (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs


Zbl 0791.35035
Full Text: DOI


[1] Dauge, Elliptic Boundary Value Problems in Corner Domains–Smoothness and Asymptotics of Solutions (1988) · Zbl 0668.35001
[2] Kondrat’ev, Trans. Moscow Math. Soc. 16 pp 227– (1967)
[3] Costabel, Symposium ’Analysis in Domains and on Manifolds with Singularities’ pp 28– (1992)
[4] Triebel, Interpolation theory. Function spaces. Differential operators (1978) · Zbl 0387.46033
[5] DOI: 10.1007/BF01199307 · Zbl 0671.58040
[6] Schmutzler, Symposium ’Analysis in Domains and on Manifolds with Singularities’, Breitenbrunn 1990 131 pp 202– (1992)
[7] Rempel, Asymptotics for Elliptic Mixed Boundary Problems (1989)
[8] DOI: 10.1080/03605308908820633 · Zbl 0719.35011
[9] Nikishkin, Moscow Univ. Math. Bull. 34 pp 53– (1979)
[10] DOI: 10.1002/mana.19921550115 · Zbl 0794.35039
[11] Maz’ya, Trans. Moscow Math. Soc. 1 pp 49– (1980)
[12] DOI: 10.1002/mana.19881380103 · Zbl 0672.35020
[13] Kufner, Some applications of weighted Sobolev spaces (1987)
[14] DOI: 10.1070/RM1983v038n02ABEH003470 · Zbl 0548.35018
[15] Kondrat’ev, Differential Equations 13 pp 1411– (1970)
[16] Costabel, C. R. Acad. Sci. Paris, Sér. I Math. 312 pp 227– (1991)
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.