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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
##### Keywords:
Cauchy problem; symbol; Levi conditions; well posed
Zbl 0713.00007