2003/04/22 by Nicolas Lerner
Mathematics · #math.AP #msc:35S05 #msc:35A05 #msc:47G30
published as Proceedings of the ICM, Beijing 2002, vol. 2, 711--720
arxiv created 2003/04/22 · arxiv updated 2009/11/30
In 1957, Hans Lewy constructed a counterexample showing that very simple and natural differential equations can fail to have local solutions. A geometric interpretation and a generalization of this counterexample were given in 1960 by L.Hörmander. In the early seventies, L.Nirenberg and F.Treves proposed a geometric condition on the principal symbol, the so-called condition (ψ), and provided strong arguments suggesting that it should be equivalent to local solvability. The necessity of condition (ψ) for solvability of pseudo-differential equations was proved by L.Hörmander in 1981. In 1994, it was proved by N.L. that condition (ψ) does not imply solvability with loss of one derivative for pseudo-differential equations, contradicting repeated claims by several authors. However in 1996, N.Dencker proved that these counterexamples were indeed solvable, but with a loss of two derivatives. We shall explore the structure of this phenomenon from both sides: on the one hand, there are first-order pseudo-differential equations satisfying condition (ψ) such that no L2loc solution can be found with some source in L2loc. On the other hand, we shall see that, for these examples, there exists a solution in the Sobolev space H-1loc.