Propagation of analytic singularities for second order Dirichlet problems. II. (English) Zbl 0534.35030

Sjöstrand, Johannes

Commun. Partial Differ. Equations 5, 187-207 (1980).

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

Keywords:

propagation; analytic singularities; second order Dirichlet problems; diffractive region; boundary bicharacteristic

Citations:

Zbl 0458.35026; Zbl 0524.35032

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