Local solvability for partial differential equations with multiple characteristics in mixed Gevrey-\(C^\infty\) spaces
The author considers the equation \[ P(w,D)u= f(x,y),\quad w=(x,y)\in \Omega,\tag{1} \] where \(\Omega\) is a neighborhood of the origin in \(\mathbb{R}^{p+q}\), and \(P(w,D)\) is a partial differential operator with analytic coefficients, of the form \(P_1(w,D_x)- P_2(w,D_y)+ Q(w,D)\). The principal part \[ P_1(w,D_x)= \sum_{|\alpha|= m}a_\alpha(w) D^\alpha_x \] is \(x\)-elliptic, the subprincipal part \[ P_2(w,D_y)= \sum_{\langle \rho,\sigma\rangle= m} b_\beta(w) D^\beta_y \] is such that its symbol has real part different from zero and imaginary part which does not change sign (here \(\sigma= (\sigma_1,\dots, \sigma_q)\), \(\sigma_j\geq 1\)). Finally, \[ Q(w,D)= \sum_{\langle\gamma,\varphi\rangle\leq m^*} c_\gamma(w) D^\gamma_w, \] \(m^*< m\), \(\varphi= (1,\dots, 1,\sigma)\in\mathbb R^{p+q}\). The main result is the following: if \(f\in C^\infty_0(\Omega)\) is Gevrey of order \(\lambda_j\) in the \(x_j\)-variable, \(1\leq j\leq p\) (or Gevrey of order \(\mu_k\) in the \(y_k\)-variable, \(1\leq k\leq q\)) with some \(\lambda_j\geq 1\), \(\mu_k\geq 1\), then the equation (1) admits a classical solution. The proof uses the machinery of pseudo-differential operators and techniques from microlocal analysis to establish the existence of a parametrix of \(P\) in the scale of Gevrey-Sobolev spaces.
scientific article; zbMATH DE number 5790117
scientific article