Nonanalytic solutions of certain linear PDEs

Abstract:

It is shown that if $P$ is a linear partial differential operator with analytic coefficients, and if $M$ is an analytic submanifold of codimensions $3$ in ${{\mathbf {R}}^n}$, which is partially characteristic with respect to $P$ and satisfies certain additional conditions, then one can find, in a neighborhood of any point of $M$, solutions of the equation $Pu = 0$ which are flat or singular precisely on $M$. The additional condition requires that a nonhomogeneous Laplace equation in two variables possesses a solution with a strong extremum at the origin. The right side of this nonhomogeneous equation is a homogeneous polynomial in two variables with coefficients being repeated Poisson brackets of the real and imaginary parts of the principal symbol of $P$.