×

zbMATH — the first resource for mathematics

Propagation of analytic singularities for second order Dirichlet problems. II. (English) Zbl 0534.35030
This is the second of three papers [part I, ibid., 41-94 (1980; Zbl 0458.35026), part III, ibid. 6, 499-567 (1981; Zbl 0524.35032) on propagation of analytic singularities for second order Dirichlet problems, with particular emphasis on the diffractive region. Through a point \(\rho_ 0\) of the diffractive region there passes the boundary bicharacteristic \(\gamma_ 0(t)\) and the \(C^{\infty}\) ray \(\alpha_ 0(t)\), defined in [R. B. Melrose and J. Sjoestrand, Commun. Pure Appl. Math. 35, 129-168 (1982)]. One considers the forward and backward half of each: for \(0<t\leq \delta_ 0,\gamma_ 1(t)=\gamma_ 0(-t),\gamma_ 2(t)=\gamma_ 0(t),\gamma_ 3(t)=\alpha_ 0(- t),\gamma_ 4(t)=\alpha_ 0(t).\) For \(\partial \Omega\) noncharacteristic for P, Pu analytic, \(u|_{\partial \Omega}\) analytic and two further technical conditions, it is shown that if \(\gamma_ 1(t)\), \(\gamma_ 2(t)\not\in WF_{ba}(u)\), \(\gamma_ 0\not\in WF_{ba}(u)\), where \(WF_{ba}(u)\) is the analytic wave front set at the boundary. In part III, other possible combinations of \(\gamma_ j(t)\not\in WF_{ba}(u)\) are settled-two affirmatively and one negative.
Reviewer: R.Johnson

MSC:
35J15 Second-order elliptic equations
35A20 Analyticity in context of PDEs
58J47 Propagation of singularities; initial value problems on manifolds
35A30 Geometric theory, characteristics, transformations in context of PDEs
58J40 Pseudodifferential and Fourier integral operators on manifolds
PDF BibTeX Cite
Full Text: DOI
References:
[1] Anderson K. G., C.R.A.S. 282 (4) pp 275– (1976)
[2] Friedlander F. G., Math. Proc. Camb. Phil.Soc. 81 pp 97– (1977) · Zbl 0356.35056
[3] Hörmander L., Linear Partial Differential Operators (1963) · Zbl 0108.09301
[4] Melrose R. B., C.P.A.M. 31 pp 593– (1978)
[5] Melrose R. B., See also :Sém 15 (1977)
[6] Rauch J., Bull. Soc. R. des Sci. de Liège pp 156– (1977)
[7] Sjöostrand J., to appear
[8] Unterberger A., Am. Inst. Fourier 21 (2) pp 85– · Zbl 0205.43104
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.