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VIDEO LIBRARY |
Sixth International Conference on Differential and Functional Differential Equations DFDE-2011
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The solvability of differential equations N. Dencker Lund University, Lund, Sweden |
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Abstract: It was a great surprise when Hans Lewy in 1957 presented a non-vanishing complex vector field that is not locally solvable. Actually, the vector field is the tangential Cauchy–Riemann operator on the boundary of a strictly pseudoconvex domain. Hörmander proved in 1960 that almost all linear partial differential equations are not locally solvable, because the necessary bracket condition is non-generic. This also has consequences for the spectral instability of non-selfadjoint semiclassical operators and the solvability of the Cauchy problem for non-linear analytic vector fields. Nirenberg and Treves formulated their famous conjecture in 1970: that condition ( The Nirenberg–Treves conjecture was finally proved in 2003. We shall present the background, the main results, and some generalizations to non-principal type equations and systems of differential equations. |