Solving pseudo-differential equations
| dc.creator | Lerner, Nicolas | |
| dc.date | 2003-04-22 | |
| dc.date.accessioned | 2026-07-07T04:57:14Z | |
| dc.date.available | 2026-07-07T04:57:14Z | |
| dc.description | 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 $L^2_{\text{loc}}$ solution can be found with some source in $L^2_{\text{loc}}$. On the other hand, we shall see that, for these examples, there exists a solution in the Sobolev space $H^{-1}_{\text{loc}}$. | |
| dc.identifier | https://arxiv.org/abs/math/0304335 | |
| dc.identifier | http://arxiv.org/abs/math/0304335 | |
| dc.identifier | Proceedings of the ICM, Beijing 2002, vol. 2, 711--720 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/67196 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35S05, 35A05, 47G30 | |
| dc.title | Solving pseudo-differential equations | |
| dc.type | text |