Nils Dencker - Microlocal and Global Analysis, ...
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Nils Dencker - Microlocal and Global Analysis, Interactions with Geometry
The Solvability of Differential Equations.
Since Hans Lewy sixty years ago presented his famous counterexample, it has been known that nonsymmetric
linear partial dierential equations are generically not solvable. For dierential operators with simple
characteristics, solvability is equivalent to the Nirenberg-Treves condition ( ). This condition involves the sign
changes of the symbol of the nonsymmetric part of the highest order terms. Condition ( ) also has consequences for
the stability of the spectrum and for nonlinear dierential equations.
In this talk, we shall consider dierential operators that have double characteristics. Examples are weakly hyperbolic
operators and parabolic operators. Then one can dene conditions corresponding to ( ) on the lower order
terms at the double characteristics. We shall show that these conditions are necessary for solvability in several cases.
A particularly interesting case are the weakly hyperbolic operators.