martedi' 28-11-2006 (16:30) - Sala delle Riunioni Nils Dencker (Universita' di Lund) : Colloquium di Dipartimento -Solvability and the Nirenberg-Treves Conjecture
In the 50's, the consensus was that all linear partial differential equations were solvable. Therefore, it came as a surprise 1957 when Hans Lewy found a non-solvable complex vector field. The vector field is a natural one, it is the Cauchy-Riemann operator on the boundary of a strictly pseudo-convex domain. The fact is that almost all linear PDE's are unsolvable, because of the Hörmander bracket condition. A rapid development in the 60's lead to the conjecture by Nirenberg and Treves in 1969: condition ($\Psi$) is necessary and sufficient for the solvability of partial differential equations of principal type. This is a condition which involves only the sign changes of the imaginary part of the highest order term of the operator along the bicharacteristics of the real part. The Nirenberg-Treves conjecture has recently been proved, see Annals of Mathematics, 163:2, 2006. We shall present the background and the ideas of the proof.
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