Semi-linear Diffraction of Conormal Waves

We study the conormal regularity of bounded solutions to semi-linear strictly hyperbolic equations on domains with diiractive boundaries: P u = f (x; u) in X; u @X = 0; u 2 L 1 loc (X): If X ? X and X is the domain of innuence of X ? we consider solutions such that singsupp(u) \ X ? \ @X = ; and further suppose that u X? is conormal with respect to a smooth characteristic hypersurface, the incoming front. For the linear equation, f 0; the singular support of u is contained in the incoming front and the reeected front obtained using the rules of geometrical optics; these two characteristic surfaces are tangent at the glancing set, the locus of points at which the incoming bicharacteristics are tangent to the boundary. We prove that in the semi-linear case the only new singularites which may occur appear on the characteristic half-cone over the glancing set. The actual conormal regularity result presented in the paper is considerably more precise. Our assumptions are best illustrated by taking for P the constant coeecient wave equation with X the product of R t and the exterior of a strictly convex obstacle. Then X ? = X \ ft < ?Tg and for the initial data one can take locally an anti-derivative of the plane wave (t ?hx; !i) with T appropriately large. The geometry of this problem in two space dimensions is shown in Figures 1 and 2. R esum e. Nous etudions la r egularit e conormale de solutions born ees d' equations semi-lin eaires strictement hyperboliques dans des domaines a bord diiractif: est le domaine d'innuence de X ? , nous consid erons des solutions u telles que singsupp(u) \ X ? \ @X = ; ; de plus nous supposons que u X? est conormale par rapport a une hypersurface caract eristique lisse, le front entrant. Dans le cas de l' equation lin eaire f 0; le support singulier de u est contenu dans la r eunion du front entrant et du front r ee echi obtenu par les lois de l'optique g eom etrique. Ces deux surfaces caract eristiques sont tangentes a l'ensemble des rayons rasants, le lieu des points o u les bicaract eristiques entrantes sont tangentes au bord. Dans le cas semi-lin eaire, nous d emontrons que si de nouvelles singularit es apparaissent alors elles apparaissent sur …