Generic properties of solutions to partial differential equations (Q1382084)

The author proves the following results: 1) Two extensions of the classical Thom transversality theorem which apply to certain function spaces, for example the spaces of solutions to certain PDEs as well as to the spaces of functions obtained by applying certain operations to spaces of solutions. The theorems proved here provide a geometric characterization of the generic properties of solutions as well as establish the stability of these properties. 2) Two classes of partial differential operators are studied. These are weighted homogeneous operators and filtered differential operators. 3) The transversality conditions together with information on the lower-order Taylor expansions allow to derive the local forms of generic solutions. This establishes their stability for several different notions of equivalence. 4) Sufficient conditions for differential operators are discussed for the transversality theorems to be applicable to the associated solution spaces. The present paper provides a beautiful account of several new results by the author in the investigation of generic properties of solutions of PDEs.