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Systèmes hyperboliques à multiplicité constante et dont le rang peut varier. (Hyperbolic systems of constant multiplicity and of variable rank). (French) Zbl 0723.35043
Recent developments in hyperbolic equations, Proc. Conf., Pisa/Italy 1987, Pitman Res. Notes Math. Ser. 183, 340-366 (1988).
Summary: [For the entire collection see Zbl 0713.00007.]
We consider the operator \[ h(x,D)\equiv a(x,D)+b(x)\equiv ID_ 0+\sum_{1\leq i\leq n}a^ i(x)D_ i+b(x) \] in a neighbourhood \(\Omega\) of 0 in \({\mathbb{R}}^{n+1}\), \(x=(x_ 0,x')\), where \(a^ i\) and b are \(m\times m\) matrices with \(C^{\infty}\) coefficients or if \(n=1\) with real analytic coefficients. We assume the characteristic roots of \[ \det (I\xi_ 0+\sum_{1\leq i\leq n}a^ i(x)\xi_ i)=0 \] are real and are of constant multiplicity.
We consider the Cauchy problem \[ h(x,D)u(x)=f(x)\in C^{\infty}(\Omega),\quad u(t,x')=g_ i(x')\in C^{\infty}(\Omega_ t). \] We first reduce the operator to the form \[ h(x,D)=ID_ 0+a_ 1(x,D')+b(x,D'), \] where \(a_ 1(x,\xi ')\) is a symbol of order 1 and nilpotent, b is a symbol of order 0, and det a(x,\(\xi\))\(=\det (I\xi_ 0+a_ 1(x,\xi '))=\xi^ m_ 0.\)
The basis of the proof is a transformation by an elliptic operator \(\Delta (x,D')\) and the theorem of Egorov. If A is the cofactor matrix, we assume \[ A=\xi^ q_ 0 {\mathcal A}(x,\xi),\text{ with } {\mathcal A}(x,0,\xi ')\not\equiv 0, \]
\[ a(x,\xi)=P(x,\xi)diag.[\xi^ p_ 0,\xi_ 0^{q_ 1},...,\xi_ 0^{q_ k},1,...,1]Q(x,\xi), \] with det Q(x,0,\(\xi\) \({}')\not\equiv 0\), \(\xi^ p_ 0,\xi_ 0^{q_ 1}...\xi_ 0^{q_ k}\) are invariant factors, and \(q=q_ 1+q_ 2+...+q_ k\), \(p\geq q_ 1\geq...\geq q_ k.\)
We define the “generalized rank” of the system to be constant if and only if det Q(x,0,\(\xi\) \({}')\neq 0.\)
We essentially assume \(C^{\infty}\) coefficients when the generalized rank is constant, and when the generalized rank is variable we assume that \(n=1\) and the coefficients are analytic.
We define new Levi conditions; we distinguish three cases, namely, \(q=0\), \(0<q<m/2\) and \(q\geq m/2\). These Levi conditions are invariant under transformations by elliptic operators \(\Delta (x,D').\)
If the generalized rank is constant, these Levi conditions are necessary and sufficient for the Cauchy problem to be well posed in each of the three following cases: \(q=0\); \(q<m/2\) and \(m\leq 4\); \(q\geq m/2\) and \(m=4.\)
In the case of variable generalized rank we discuss several cases if \(m\leq 4\). In these cases also the Levi conditions are still necessary and sufficient. We use here a method of differential desingularization.
MSC:
35L45 Initial value problems for first-order hyperbolic systems
35S10 Initial value problems for PDEs with pseudodifferential operators
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
Citations:
Zbl 0713.00007