Type: Article
Publication Date: 1986-06-01
Citations: 61
DOI: https://doi.org/10.21099/tkbjm/1496160384
In the study of hyperbolic partial differential operators, it is important to in- vestigate properties of the characteristic roots.Bronshtein [2] proved the Lipschitz continuity of the characteristic roots of hyperbolic operators with variable coeffi- cients, and he studied the hyperbolic Cauchy problem in Gevrey classes (see [3]).Ohya and Tarama [7] extended the results in [2] and, also, studied the Cauchy problem.In this paper we shall give an alternative proof of Bronshtein's results, which seems to be simpler.Also, we shall prove the inner semi-continuity of the cones defined for the localization polynomials of hyperbolic operators (see Theorem 3 below).In studying singularities of solutions the inner semi-continuity of the cones plays a key role (see [8], [9], [10]).We note that our method can be appli- cable to the mixed problem.Let $p(t, x, y)=t^{m}+\Sigma_{j=1}^{m}a_{j}(x, y)t^{m-j}$ be a polynomial in $t$ , where the $a_{j}(x, y)$ are defined for $x=(x_{1}, \cdots, x_{n})\in X$ and $y\in Y,$ $X$ is an open convex subset of $R^{n}$ and $Y$ is a compact Hausdorff topological space.We assume that (A-1) $p(t, x, y)\neq 0$ if ${\rm Im} t\neq 0$ and $(x, y)\in X\times Y$ , (A-2) $\partial_{x}^{a}a_{j}(x, y)(|\alpha|\leqq k, 1\leqq j\leqq m)$ are continuous and there are $C>0$ and $\delta$ with $0<\delta\leqq 1$ such thatand $y\in Y$ , where $k$ is a nonnegative integer and $\partial_{x}^{a}=$ $(\partial/\partial x_{1})^{\alpha_{1}}\cdots(\partial/\partial x_{n})^{\alpha}n$ THEOREM 1. Assume that (A-1) and (A-2) are satisfied.Then, for any open subset $U$ of $X$ with $U\subset X$ there is $C=C(U)>0$ such that $|\lambda_{j}(x, y)-\lambda_{j}(x^{\prime}, y)|\leqq C|x-x^{\prime}|^{r}$ for $1\leqq j\leqq m,$ $x,$ $x^{\prime}\in U$ and $y\in Y$ , where $p(t, x, y)=\prod_{j\subset 1}^{m}(t-\lambda_{j}(x, y)),$ $\lambda_{1}(x, y)\leqq\lambda_{2}(x, y)\leqq\cdots\leqq\lambda_{m}(x, y)$ , and $r=\min(1, (k+\delta)$