Strongly hyperbolic systems of maximal rank
Let \(L\) be a first-order differential operator defined in an open set \(\Omega\) in \(\mathbb{R}^{n+1}\) \[ L(x, D)= D_0I_m+ \sum^n_{j= 1} A_j(x) D_j, \] where \(A_j(x)\) are real analytic \(m\times m\) matrices defined in \(\Omega\) and \(x= (x_0, x')= (x_0,x_1,\dots, x_n)\), \(\xi= (\xi_0,\xi')= (\xi_0,\xi_1,\dots, \xi_n)\). Let us denote by \(h(x,\xi)\) and \(M(x,\chi)\) the determinant and the cofactor matrix of \(L(x,\xi)\) respectively. Let \(\Sigma= \{z= (x,\xi)| h(z)=\cdots= d^{m-1} h(z)= 0\}\) be the set of characteristics of order \(m\) of \(h\). We assume that \(\Sigma\) is a real analytic manifold near a reference point \(\widehat z= (\widehat x,\widehat\xi)\). Without restrictions we may suppose that \(0\in\Omega\) and \(\widehat x=0\). Let \(\Sigma\) be given by \[ \phi_0(x,\xi)= \xi_0= 0,\quad \phi_j(x,\xi')= 0,\quad 1\leq j\leq k, \] where \(\phi_j(x,\xi')\) are real analytic, homogeneous of degree \(0\) in \(\xi'\) with linearly independent differentials at \(\widehat z\). We assume that \(L(x,\xi)\) satisfies a necessary condition for strong hyperbolicity, that is \(M(x,\xi)\) vanishes of order \(m-2\) on \(\Sigma\) which implies in particular \((L|_\Sigma)^2= 0\) where \(L|_\Sigma\) is the restriction of \(L\) to \(\Sigma\). Thus we have \(0\leq \text{rank}(L|_\Sigma)\leq [m/2]\). In our previous papers [J. Funct. Anal. 132, No. 2, 251-272 (1995; Zbl 0844.35063); Osaka J. Math. 32, No. 1, 41-69 (1995; Zbl 0830.35073)], we studied the extreme case that \(\text{rank}(L|_\Sigma)= 0\), the case closest to symmetric systems. In this note, we study the other extreme case when \(\text{rank}(L|_\Sigma)= [m/2]\), which is, in a sense, farthest away from symmetric systems. Our aim is to show that if the localization \(h_z\) of \(h(x,\xi)\) at \(z\in\Sigma\), the first non-trivial term in the Taylor expansion of \(h\) at \(z\) which is a polynomial on \(T_z(T^*\Omega)/T_z\Sigma\), is strictly hyperbolic and the propagation cone of \(h_z\) is transversal to \(\Sigma\) at every \(z\in\Sigma\) then \(L(x,D)\) is strongly hyperbolic (Theorem 1.2). Here the propagation cone is defined as the dual cone of the hyperbolic cone of \(h_z\) with respect to the canonical symplectic structure on \(T_z(T^*\Omega)\).
scientific article; zbMATH DE number 1195596