Nils Dencker: The Solvability of Differential Operators

Abstract:

Sixty years ago Hans Lewy presented his famous counterexample, a tangential Cauchy-Riemann operator on the boundary of a strictly pseudoconvex domain that is not solvable. Hˆrmander then proved in 1960 that linear partial differential operators are generically not solvable. For differential operators with simple characteristics, local solvability is equivalent to the Nirenberg-Treves condition \((\Psi)\). This condition involves the non-symmetric part of the highest order term and has important consequences for the spectral stability of the operator.

In this talk, we shall consider differential operators that have multiple characteristics. Then the solvability may depend on the lower order terms, and one can define conditions corresponding to \((\Psi)\) on these terms. We shall show that this condition is necessary for solvability in several cases.