Reduction of orders in boundary value problems without transmission property. (English) Zbl 1050.58019

Harutjunjan, Gohar; Schulze, Bert-Wolfgang

J. Math. Soc. Japan 56, No. 1, 65-85 (2004).

Authors’ summary: Boundary value problems for elliptic pseudo-differential operators without the transmission property at the boundary appear in parametrices of mixed elliptic problems such as the Zaremba problem. The paper is devoted to the construction of order reductions in boundary value problems on a manifold with smooth boundary which play a crucial role in calculating the number of additional interface conditions. Moreover, the authors show that the order reducing symbols have the Volterra property and are parabolic of anisotropy 1. The results are then extended to order reducing holomorphic operator-valued Mellin symbols for reducing orders in weighted Sobolev spaces on a manifold with conical singularities on the boundary.

MSC:

58J40	Pseudodifferential and Fourier integral operators on manifolds
35J40	Boundary value problems for higher-order elliptic equations
35S15	Boundary value problems for PDEs with pseudodifferential operators
35K25	Higher-order parabolic equations
58J32	Boundary value problems on manifolds

Keywords:

boundary value problems; elliptic operators; order reduction; Volterra symbols