Some example of locally solvable anisotropic partial differential operators (Q2704051)

The author studies the local solvability of some linear and nonlinear partial differential equations in the frame of the anisotropic Gevrey classes \(G^\sigma\), \(\sigma= (\sigma_1,\dots, \sigma_n)\). We recall that \(f\in G^\sigma\) means that the following estimates are valid in compact sets: NEWLINE\[NEWLINE|D^\lambda f(x)|\leq C^{|\lambda|+1} (\lambda_1!)^{\sigma_1}\dots (\lambda_n!)^{\sigma_n}.NEWLINE\]NEWLINE As a representative example we quote the following, for given integers \(m\), \(q\) with \(m\) odd, \(q\) even, \(q< m\): NEWLINE\[NEWLINEP(x,D)u+ F(x,D_x^\lambda u)= f(x), \quad\lambda= (\lambda',\lambda_n), \quad|\lambda'|m/q+\lambda_n< m,NEWLINE\]NEWLINE where \(P(x,D)= D_{x_n}^m- r_q(x,D')\), \(r_q(x,D')= \sum_{|\lambda'|=q} a_{\lambda'}(x) D_{x'}^{\lambda'}\). Assuming \(\operatorname {Re} r_q(x,\eta')> 0\) for \(\eta'\neq 0\) and \(\operatorname {Im} r_q(x,\eta')\geq 0\) or otherwise \(\leq 0\), the author is able to conclude local solvability for \(f\in G^\sigma\), with \(\sigma\) depending on \(m\) and \(q\). The result is new both in the linear and in the semilinear case. For similar results of Gevrey solvability see \textit{M. Mascarello} and \textit{L. Rodino} [Partial differential equations with multiple characteristics, Academie Verlag Berlin (1997; Zbl 0888.35001)]. We emphasize that Gevrey solvable operators are not \(C^\infty\) solvable, in general.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00030].