Singularities of boundary value problems. II. (English) Zbl 0546.35083

The authors generalize results which they obtained in part I [ibid. 31, 593-617 (1978; Zbl 0368.35020)]. Here they consider operators of the following kind: \[ P=D^ 2_ x+2A(x,y,D_ y)D_ x+B(x,y,D_ y) \] where A, B are pseudodifferential operators on \({\mathbb{R}}^ n_ y\), which depend smoothly on \(x\in {\bar {\mathbb{R}}}_+\), and with real principal symbols of order 1 and 2. They consider the following situation: (*) \(Pu\in C^{\infty}\) in \(x\geq 0\), \([D_ x+A(0,y,D_ y)]u+L(y,D_ y)u\in C^{\infty}\) in \(x=0\), where L is a first-order pseudodifferential operator (in part I, they considered a differential operator, with the Dirichlet (or Neumann) boundary condition). The article under review is very terse and many of the results are very concise and not easy to state in a review. Let us just point to the following. Assume that the differential dp of the principal symbol of P, and the canonical one-form are independent on \(\Sigma =\{p=0\}\), and that \(dr_ 0=d\{b(0,y,\eta)-a^ 2(0,y,\eta)\}\) and the one-form \(\Sigma\eta_ jdy_ j\) are independent on \(\{r_ 0=0\}\); then (in the case of the Dirichlet condition), if u satisfies (*), the broken wave front set \(WF_ b(u)\) of u is a union of maximally extended generalised bicharacteristics.


35S15 Boundary value problems for PDEs with pseudodifferential operators
47Gxx Integral, integro-differential, and pseudodifferential operators
35A20 Analyticity in context of PDEs


Zbl 0368.35020
Full Text: DOI


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