zbMATH — the first resource for mathematics

On some global aspects of the theory of partial differential equations on manifolds with singularities. (English) Zbl 0898.35120
Gohberg, I. (ed.) et al., Differential and integral operators. Vol. I. Proceedings of the international workshop on operator theory and applications, IWOTA 95, Regensburg, Germany, July 31-August 4, 1995. Basel: Birkäuser. Oper. Theory, Adv. Appl. 102, 287-305 (1998).
The starting point for the paper is the following remark, concerning elliptic partial differential equations on manifolds with conical singularities: the asymptotic expansions of the solutions at different singular points are related to each other. This is tested by the authors on some simple models. Precise results are then obtained for two-dimensional equations; the asymptotics near each singular point have in this case the form \[ u(r,\varphi) \sim \sum_k r^{S_k} \sum^{m_k}_{j=0} a_{kj} (\varphi) \ln^jr, \] where \(r\) is the radial variable and \(\varphi\) is the angular variable. Relations between expansions at different points are explained in terms of propagation of singularities in the complex domain, along the degeneration set of the operator. The case of dimension \(n>2\) is also discussed.
For the entire collection see [Zbl 0883.00019].
Reviewer: L.Rodino (Torino)

35S05 Pseudodifferential operators as generalizations of partial differential operators
58J47 Propagation of singularities; initial value problems on manifolds
35A20 Analyticity in context of PDEs