Nonlinear microlocal analysis (Q2759775)

The authors survey some recent contributions in the field of nonlinear microlocal analysis concerning solvability and regularity for semilinear partial differential equations of the form NEWLINE\[NEWLINEP(x,D)v+ F(x,D^\alpha v)_{|\alpha|\leq m-1}=0 \tag{*}NEWLINE\]NEWLINE with linear part \(P(x,D)= \sum_{|\alpha |= m}a_\alpha (x)D^\alpha\) having analytic coefficients and multiple characteristics, and \(F\) is analytic in all arguments. The main result is that, if the principal part \(p(x,\xi) \equiv e_{m-k}a_1 (x,\xi)\cdots a_k(x, \xi)\) for a conic neighborhood \(\Gamma\) of \((x_0, \xi_0)\in T^*/R^{2n}\) with \(p(x_0, \xi_0)=0\) in \(\Gamma\), where \(e_{m-k}\neq 0\) and \(a_j(x,\xi)\) are first order analytic symbols of nondegenerate principal type satisfying that \(\text{Im} a_j(x,\xi)\) has the same sign in \(\Gamma\) and for all \(j\), then the equation (*) can be solved in Gevrey class of index \(1<\sigma <k/(k-1)\) microlocally at \((x_0,\xi_0)\).NEWLINENEWLINEFor the entire collection see [Zbl 0969.00056].