The hyperbolic Cauchy problem.

*(English)*Zbl 0762.35002
Lecture Notes in Mathematics. 1505. Berlin etc.: Springer-Verlag. vii, 168 p. (1991).

This is a collection of highly technical lectures given by the authors on the question of existence and uniqueness of solution of the Cauchy problem for hyperbolic operators with multiple characteristics. Part I considers the Cauchy problem for hyperbolic operators in Gevrey classes while Part II treats the hyperbolic operators in \(C^ \infty\) classes.

To overcome the difficulty which arises from the loss of derivatives of solutions in the theory of partial differential operators, the authors introduce the so-called Fourier integral operators with complex-valued phase function \(\varphi(x,\xi)=x\cdot\xi-i\Lambda(x,\xi)\), where \(\Lambda(x,\xi)\) is a real-valued function in a suitable class. For the Cauchy problem in Gevrey class \(E^{\{\kappa\}}(\kappa\geq 1)\), \(\Lambda(x,\xi)\) is chosen to satisfy \(\Lambda(x,\xi)=O(|\xi|^{1/\kappa})\) while in the \(C^ \infty\) class it is chosen to satisfy \(\Lambda(x,\xi)=O(\log|\xi|)\), as \(|\xi|\to\infty\). For \(\varphi=x\cdot\xi-i\Lambda(x,\xi)\), the authors define the Fourier integral operator \(I_ \varphi(x,D)u(x)=\int e^{i\varphi(x,\xi)}\hat u\hat d\xi\) for \(u\in C_ 0^ \infty(\mathbb{R}^ n)\times E^{\{\kappa\}}(\mathbb{R}^ n)\) (resp. \(C_ 0^ \infty(\mathbb{R}^ n))\), where \(u\) stands for the Fourier transform of \(u\) and \(d\xi=(2\pi)^{-n}d\xi\). If \(\exp\Lambda(x,\xi)\) is chosen to compensate for the loss of derivatives of solutions, then it can be expected that operators with loss of infinite (resp. finite) derivatives in Gevrey classes (resp. \(C^ \infty\) classes) will be transformed to “good” operators by conjugation with such a Fourier integral operator. For a pseudo-differential operator \(p(x,D)\), the authors define \(p_ \Lambda(x,D)=I^ R_{-\varphi}(x,D)p(x,D)I_ \varphi(x,D)\), where \(I^ R_{-\varphi}(x,D)\) denotes the reverse operator of \(I_{- \varphi}(x,D)\). The symbol \(p_ \Lambda(x,\xi)\) is thus given by \(p_ \Lambda(x,\xi)\sim p(x+i\Lambda_ \xi,\xi-i\Lambda_ x)+\cdots\), where \(p(x+iy,\xi-i\eta)\) is an almost analytic extension of \(p(x,\xi)\). The task of choosing a suitable \(\Lambda(x,\xi)\) so that one can derive a priori estimates for \(p_ \Lambda(x,D)\) leads to consideration of so- called special time functions, the properties of which are closely connected with the geometry of the bicharacteristics and the multiple characteristic set of \(p(x,\xi)\).

To overcome the difficulty which arises from the loss of derivatives of solutions in the theory of partial differential operators, the authors introduce the so-called Fourier integral operators with complex-valued phase function \(\varphi(x,\xi)=x\cdot\xi-i\Lambda(x,\xi)\), where \(\Lambda(x,\xi)\) is a real-valued function in a suitable class. For the Cauchy problem in Gevrey class \(E^{\{\kappa\}}(\kappa\geq 1)\), \(\Lambda(x,\xi)\) is chosen to satisfy \(\Lambda(x,\xi)=O(|\xi|^{1/\kappa})\) while in the \(C^ \infty\) class it is chosen to satisfy \(\Lambda(x,\xi)=O(\log|\xi|)\), as \(|\xi|\to\infty\). For \(\varphi=x\cdot\xi-i\Lambda(x,\xi)\), the authors define the Fourier integral operator \(I_ \varphi(x,D)u(x)=\int e^{i\varphi(x,\xi)}\hat u\hat d\xi\) for \(u\in C_ 0^ \infty(\mathbb{R}^ n)\times E^{\{\kappa\}}(\mathbb{R}^ n)\) (resp. \(C_ 0^ \infty(\mathbb{R}^ n))\), where \(u\) stands for the Fourier transform of \(u\) and \(d\xi=(2\pi)^{-n}d\xi\). If \(\exp\Lambda(x,\xi)\) is chosen to compensate for the loss of derivatives of solutions, then it can be expected that operators with loss of infinite (resp. finite) derivatives in Gevrey classes (resp. \(C^ \infty\) classes) will be transformed to “good” operators by conjugation with such a Fourier integral operator. For a pseudo-differential operator \(p(x,D)\), the authors define \(p_ \Lambda(x,D)=I^ R_{-\varphi}(x,D)p(x,D)I_ \varphi(x,D)\), where \(I^ R_{-\varphi}(x,D)\) denotes the reverse operator of \(I_{- \varphi}(x,D)\). The symbol \(p_ \Lambda(x,\xi)\) is thus given by \(p_ \Lambda(x,\xi)\sim p(x+i\Lambda_ \xi,\xi-i\Lambda_ x)+\cdots\), where \(p(x+iy,\xi-i\eta)\) is an almost analytic extension of \(p(x,\xi)\). The task of choosing a suitable \(\Lambda(x,\xi)\) so that one can derive a priori estimates for \(p_ \Lambda(x,D)\) leads to consideration of so- called special time functions, the properties of which are closely connected with the geometry of the bicharacteristics and the multiple characteristic set of \(p(x,\xi)\).

Reviewer: E.C.Young (Tallahassee)