Nils Dencker: The solvability and range of differential equations

Hörmander proved in his thesis 1955 that symmetric linear differential equations having simple characteristics are locally solvable. It was at that time expected that this was true for all linear partial differential equations.

Therefore it was a great surprise in 1957 when Hans Lewy presented a complex vector field that is not solvable anywhere. Actually, the vector field is the tangential Cauchy-Riemann equation on the boundary of a strictly pseudoconvex domain. Hˆrmander then proved in 1960 that in fact almost all linear partial differential equations are not solvable. For nonsolvable equations the range is of first category, and Hörmander proved in 1963 that nonsolvable complex vector fields are determined by their range, up to right multiplication by functions.

We shall generalize this to nonsolvable systems of differential equations of constant characteristics, including scalar equations. We show that the ranges of these equations determine the coefficients up to infinite order at minimal bicharacteristics, up to right composition by differential equations. The minimal bicharacteristics are the smallest sets on which the equation is not solvable. This is joint work with Jens Wittsten.