On the critical index of Gevrey solvability for some linear partial differential equations (Q2756121)

This paper deals with the precise estimates of the critical index of Gevrey solvability to some classes of linear partial differential equations (p.d.e.) with multiple characteristics. The corresponding Gevrey classes of index \(\sigma\geq 1\) are denoted by \(G^\sigma\). At first the author considers the following class of quasihomogeneous p.d.e. in \(\mathbb{R}^2\): NEWLINE\[NEWLINEQ=D_t^2 +t^{2k}D_x^{2q}+ bt^{k-1} D_x^q,\;k-\text{even}. \tag{1}NEWLINE\]NEWLINE The critical Gevrey index for non-solvability of (1) is \((k+q)/(k+1)\geq 1\).NEWLINENEWLINENEWLINEThe second result concerns the \(\sigma\)-solvability for a p.d.e. (P) with involutive characteristics of codimension 1 (i.e. \(\tau=0)\) and elliptic subprincipal symbol with imaginary part conserving its sign. It is proved then that (P) is \(\sigma\)-locally solvable for \(1<\sigma <{m\over m-2}\), \(m\geq 3\) and \(1<\sigma \leq\infty\) if \(m=2\).