On the solvability of linear partial differential equations
The authors introduce a purely algebraic method for solvability of partial differential equations, suitable also for treating \({\mathcal D}_X\)-modules, which are not hyperbolic, nor simple-characteristic. The main result they prove is as follows. Let \(N=\{x_1=0\}\) be a closed submanifold of the real analytic manifold \(M,\) and denote by \(Y\subset X\) a complexification of \(N\subset M.\) If \({\mathcal M}\) is a coherent \({\mathcal D}_X\)-module for which \(Y\) is non-characteristic, and if \({\mathcal M}\) is micro-hyperbolic in the directions \(\pm dx_1\in T_N^*M\setminus 0\) on \(N\times_M(T_M^*X\setminus 0),\) then \(\text{Ext}_{{\mathcal D}_X}^j({\mathcal M},{\mathcal B}_M)\bigl|_N\simeq 0\) for \(j>\text{proj.dim}{\mathcal M}_Y,\) where \({\mathcal B}_M\) denotes the sheaf of Sato's hyperfunctions on \(M\) and \({\mathcal M}_Y\) the system induced by \({\mathcal M}\) on \(Y.\) Some variants of the main theorem and applications, especially to the case \(P=EQ+(\)lower-order-terms), where \(E\) is elliptic and \(Q\) is semihyperbolic (resp. hyperbolic) in the direction \(dx_1\) (resp. \(\pm dx_1),\) are then given.
scientific article; zbMATH DE number 952360