Solving Pseudo-Differential Equations

摘要

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.Hormander. 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.Hormander in 1981. The sufficiency of condition (ψ) for solvability of differential equations was proved by R.Beals and C.Fefferman in 1973. For differential equations in any dimension and for pseudo-differential equations in two dimensions, it was shown more precisely that (ψ) implies solvability with a loss of one derivative with respect to the elliptic case: for instance, for a complex vector field X satisfying (ψ), f ∈L_(loc)~2, the equation Xu = f has a solution u ∈ L_(loc)~2. 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_(loc)~2 solution can be found with some source in L_(loc)~2. On the other hand, we shall see that, for these examples, there exists a solution in the Sobolev space H_(loc)~(-1). The sufficiency of conditions (ψ) for solvability of pseudo-differential equations in three or more dimensions is still an open problem. In 2001, N.Dencker announced that he has proved that condition (ψ) implies solvability (with a loss of two derivatives), settling the Nirenberg-Treves conjecture. Although his paper contains several bright and new ideas, it is the opinion of the author of these lines that a number of points in his article need clarification.