# zbMATH — the first resource for mathematics

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.

##### MSC:
 35S15 Boundary value problems for PDEs with pseudodifferential operators 47Gxx Integral, integro-differential, and pseudodifferential operators 35A20 Analyticity in context of PDEs
Full Text:
##### References:
 [1] Andersson, Invent. Math. 41 pp 197– (1977) [2] Beals, Comm. Pure Appl. Math. 27 pp 1– (1974) [3] Beals, Comm. Pure Appl. Math. 27 pp 161– (1974) [4] Hörmander, Enseignement Math. 17 pp 99– (1971) [5] Imai, Hokkaido Math. J. 7 pp 339– (1978) · Zbl 0409.35054 [6] Gårding, C. R. Acad. Sci. Paris 285 (1977) [7] Rectification 285 pp 1199– (1978) · JFM 25.0143.02 [8] and , Scattering Theory, Academic Press, New York, 1967. [9] Melrose, Invent. Math. 37 pp 165– (1976) [10] Melrose, Comm. in P.D.E. 3 pp 1– (1978) [11] Melrose, Duke Math. J. 46 pp 43– (1979) [12] Melrose, Comm. Pure Appl. Math. 31 pp 593– (1978) [13] Miyatake, Japan J. Math. 1 pp 111– (1975) [14] Morawetz, Comm. Pure Appl. Math. 30 pp 447– (1977) [15] Ralston, Comm. Pure Appl. Math. 22 pp 807– (1969) [16] Taylor, Comm. Pure Appl. Math. 29 pp 1– (1978) [17] Guillot, C. R. Acad. Sci. Paris 292 pp 43– (1981)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.