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The hyperbolic region for hyperbolic boundary value problems

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Coulombel, J.-F. Osaka J. Math. 48 (2011), 457–469 THE HYPERBOLIC REGION FOR HYPERBOLIC BOUNDARY VALUE PROBLEMS JEAN-FRANÇOIS COULOMBEL (Received October 7, 2008, revised January 6, 2010) Abstract The well-posedness of hyperbolic initial boundary value problems is linked to the occurrence of zeros of the so-called Lopatinskiı˘ determinant. For an important class of problems, the Lopatinskiı˘ determinant vanishes in the hyperbolic region of the frequency domain and nowhere else. In this paper, we give a criterion that ensures that the hyperbolic region coincides with the projection of the forward cone. We give some examples of strictly hyperbolic operators that show that our criterion is sharp. 1. Introduction In this paper, we consider initial boundary value problems for hyperbolic systems. Such problems read: (1) 8 < : Lu WD �t u C d X jD1 A j �x j u D F(t , x), (t , x) 2 RC � Rd C , Bu(t , y, 0) D g(t , y), (t , y) 2 RC � Rd�1, u(0, x) D f (x), x 2 Rd C , where the spatial domain is the half-space Rd C D fx 2 Rd , xd > 0g and the notation x D (y, xd ) is used. The A j ’s are N � N real matrices, and B is a p � N real matrix. We always assume d � 2 in what follows. The well-posedness of (1) can be characterized with the help of a complex valued function 1, that is known as the Lopatinskiı˘ determinant and that depends on the vari- ables (z, �), z 2 C with Im z � 0 and � 2 Rd�1. We refer to the original articles [5, 9, 10] as well as to the book [2, Chapter 4] for a detailed description of the theory. The function 1 can be chosen to be positively homogeneous of degree 0 with respect to the variables (z, �). If 1 does not vanish on the closed half-sphere fIm z � 0, jzj2C j�j 2 D 1g, then (1) is strongly well-posed, meaning that source terms in L2 give rise to a unique solution u in L2 that depends continuously on the data. When 1 vanishes in the open half-sphere fIm z < 0, jzj2 C j�j2 D 1g, (1) is ill-posed. In [1],

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