##
**Wave propagation in dihedra.
(Propagation des ondes dans les dièdres.)**
*(French)*
Zbl 0845.35061

The wave equation and the problem of the propagation of analytic singularities in domains with conical and angular points is of constant interest during the last two decades. After the fundamental papers of J. Sjöstrand [Commun. Partial Differ. Equ. 5, 41-94 (1980; Zbl 0458.35026)], [Astérisque 95, 1-166 (1982; Zbl 0524.35007)], [Math. Ann. 254, 211-256 (1980; Zbl 0459.35007)] many authors used the developed sophisticated technique for the study of the mentioned problem in different special cases.

In the Memoir under review the wave equation of finite energy in a domain \(\Omega\) of \(\mathbb{R}^d\), with Dirichlet boundary value conditions, near a codimension 2 corner of the boundary (called “dièdre a faces courbées”) is considered. In fact, some generalizations of the previous joint paper of the author with P. Gerard [J. Am. Math. Soc. 6, No. 2, 341-424 (1993; Zbl 0779.35063)] are given. Let us recall that in the cited joint paper the asymptotics on the diffracting cone for the associated solution to a conormal incoming wave was developed in the case \(d= 2\). It is to remark that in the case of arbitrary dimension \(d\) the situation is more complicated. New tangent rayons appear along the edge of the angular corner, and the propagation on these rayons is to be studied. So the principal success is that the developed theory is deliberated of any restriction of the type “incoming conormal singularities” and that the treatments of the interior \(\Omega^i\) and the exterior \(\Omega^e\) of the corner are made conjointly. The realization of all this is obtained with the help of different innovations of technical character.

More precisely, the study is based on some geometric concepts (different cotangent spaces, especially the characteristic manifold \(\Sigma_b\) constructed for the considered boundary value problem) and the notion of an “analytic wave front \(SS_b(u)\) up to the boundary” for the solution \(u\in H^1_{\text{loc}}\) of the Dirichlet problem \(Pu= 0\), \(u|_{\partial M}= 0\) with \(M= R_t\times \Omega\), or \(M= \Omega^i\), or \(M= \Omega^e\) as a subset of \(\dot T^*_b M\). The proposed construction of \(SS_b(u)\) uses tangent FBI transforms and some a priori estimation obtained by the author. Developing some microlocalization and some elliptic estimations for pseudo-differential equations with values in a chain of Banach spaces, the author carefully prepares the statement and the proofs of his main results for the equation \(Pu= 0\) with \(u\in H^1_0\) and \(u|_{\partial M}= 0\). Roughly speaking these are: the independence of \(SS_b(u)\) of the choice of the coordinates (the choice of the space of normal regularity), the elliptic regularity \((SS_b(u)\subset \Sigma_b)\) and the hyperbolic reflection on the edge. The above mentioned propagation on the tangent to the edge rayons is developed using the Sjöstrand microhyperbolic deformation strategy (the above cited paper in Math. Ann.). Introducing the notion of glancing region \(\mathcal G\), the author obtains estimations near the strata \(S^2_{\mathcal G}\) of \(\mathcal G\). All previous results are applied in the last chapter for the study of the propagation of the analytic singularities:

Theorem. For a solution \(u\in H^1_0\) of \(Pu= 0\), \(SS_b(u)\) is closed in \(\Sigma_b\), and if \(\rho_0\in SS_b(u)\) there exists a maximal exiting rayon from \(\rho_0\) in \(SS_b(u)\).

Author’s results on the second microlocalization are recalled in an appendix. In general, the Memoir is written with maximal precision, difficult to be refered in short, but it is quite good as an introduction in the subject for a pedantic reader.

In the Memoir under review the wave equation of finite energy in a domain \(\Omega\) of \(\mathbb{R}^d\), with Dirichlet boundary value conditions, near a codimension 2 corner of the boundary (called “dièdre a faces courbées”) is considered. In fact, some generalizations of the previous joint paper of the author with P. Gerard [J. Am. Math. Soc. 6, No. 2, 341-424 (1993; Zbl 0779.35063)] are given. Let us recall that in the cited joint paper the asymptotics on the diffracting cone for the associated solution to a conormal incoming wave was developed in the case \(d= 2\). It is to remark that in the case of arbitrary dimension \(d\) the situation is more complicated. New tangent rayons appear along the edge of the angular corner, and the propagation on these rayons is to be studied. So the principal success is that the developed theory is deliberated of any restriction of the type “incoming conormal singularities” and that the treatments of the interior \(\Omega^i\) and the exterior \(\Omega^e\) of the corner are made conjointly. The realization of all this is obtained with the help of different innovations of technical character.

More precisely, the study is based on some geometric concepts (different cotangent spaces, especially the characteristic manifold \(\Sigma_b\) constructed for the considered boundary value problem) and the notion of an “analytic wave front \(SS_b(u)\) up to the boundary” for the solution \(u\in H^1_{\text{loc}}\) of the Dirichlet problem \(Pu= 0\), \(u|_{\partial M}= 0\) with \(M= R_t\times \Omega\), or \(M= \Omega^i\), or \(M= \Omega^e\) as a subset of \(\dot T^*_b M\). The proposed construction of \(SS_b(u)\) uses tangent FBI transforms and some a priori estimation obtained by the author. Developing some microlocalization and some elliptic estimations for pseudo-differential equations with values in a chain of Banach spaces, the author carefully prepares the statement and the proofs of his main results for the equation \(Pu= 0\) with \(u\in H^1_0\) and \(u|_{\partial M}= 0\). Roughly speaking these are: the independence of \(SS_b(u)\) of the choice of the coordinates (the choice of the space of normal regularity), the elliptic regularity \((SS_b(u)\subset \Sigma_b)\) and the hyperbolic reflection on the edge. The above mentioned propagation on the tangent to the edge rayons is developed using the Sjöstrand microhyperbolic deformation strategy (the above cited paper in Math. Ann.). Introducing the notion of glancing region \(\mathcal G\), the author obtains estimations near the strata \(S^2_{\mathcal G}\) of \(\mathcal G\). All previous results are applied in the last chapter for the study of the propagation of the analytic singularities:

Theorem. For a solution \(u\in H^1_0\) of \(Pu= 0\), \(SS_b(u)\) is closed in \(\Sigma_b\), and if \(\rho_0\in SS_b(u)\) there exists a maximal exiting rayon from \(\rho_0\) in \(SS_b(u)\).

Author’s results on the second microlocalization are recalled in an appendix. In general, the Memoir is written with maximal precision, difficult to be refered in short, but it is quite good as an introduction in the subject for a pedantic reader.

Reviewer: S.Dimiev (Monastir)

### Keywords:

wave equation of finite energy; Dirichlet boundary conditions; hyperbolic reflection; propagation of analytic singularities; domains with conical and angular points
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\textit{G. Lebeau}, Mém. Soc. Math. Fr., Nouv. Sér. 60, 124 p. (1995; Zbl 0845.35061)

### References:

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