## 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

### Citations:

Zbl 0458.35026; Zbl 0524.35032
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### References:

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