PDE Seminar: Nils Dencker, "The Solvability of Differential Equations"
Since Hans Lewy sixty years ago presented his famous counterexample, it has been known that almost all nonsymmetric linear partial differential equations are not solvable. For differential equations with simple characteristics, solvability is equivalent to the Nirenberg-Treves condition (PSI). This condition involves the nonsymmetric part of the highest order terms.
In this talk, we shall consider differential operators that have double characteristics. Then the solvability may depend on the lower order terms, and one can define a condition corresponding to (PSI) on these terms. We shall show that this condition is necessary for solvability in several cases.