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Asymptotic solutions of hyperbolic boundary value problems with diffraction. (English) Zbl 0601.35113

Advances in microlocal analysis, Proc, NATO Adv. Study Inst., Castelvecchio-Pascoli (Lucca)/Italy 1985, NATO ASI Ser., Ser. C 168, 165-202 (1986).
[For the entire collection see Zbl 0583.00014.]
This paper concerns the propagation of analytic singularities in boundary value problems. The author is mainly interested in the problem \[ (1)\quad (-\Delta +D^ 2_ t)u\in A({\mathbb{R}}\times \Omega);\quad u|_{{\mathbb{R}}\times \partial \Omega}\in A({\mathbb{R}}\times \partial \Omega), \] where \(\Omega\) is an open subset of \({\mathbb{R}}^{n-1}\) with analytic boundary. It is known that the problem (1) can be reduced locally to \[ (2)\quad Pu\in A(M),\quad u|_{\partial M}\in A(\partial M) \] where \(P(x,D)=D^ 2_{x_ n}+R(x,D_ x)\), \(M=u'\times [0,a)\), \(a>0\) and u’ is an open neighbourhood of 0 in \({\mathbb{R}}^{n-1}\). If \(p(x,\xi)=\xi^ 2_ n+r(x,\xi ')\) denotes the principal symbol of P, \(r_ 0(x',\xi ')=r(x',0,\xi ')\), the author considers (2) in the so called diffractive region \(G_+\), where \(r_ 0(x',\xi ')=0\). \((\partial /\partial x_ n)r(x',0,\xi ')<0\). The author constructs asymptotic solutions to the diffractive problem which are singular on the bicharacteristics of p but do not propagate singularities at the boundary.
Reviewer: M.Schneider

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
35C20 Asymptotic expansions of solutions to PDEs
58J45 Hyperbolic equations on manifolds
47Gxx Integral, integro-differential, and pseudodifferential operators

Citations:

Zbl 0583.00014