Question

This question is a follow up of the following answer I posted recently:

http://mathoverflow.net/questions/68680/counterexamples-in-pde/97250#97250

Is there a second order partial differential operator with real coefficients which are not solvable in Lewy sense? I am also not aware of first and third order operators but I thought second order operator is a natural question as we may be able to apply some conjugations of known first order non-solvable Lewy like operators with complex coefficients. In this direction, Mizohata's example $\frac{\partial }{\partial x}+ix\frac{\partial }{\partial y}$ was not helpful as it is giving me Grushin like operators whose solvability is proved.

Answer 1

The operator $\frac{\partial}{\partial t}+t\Delta_x$ is not locally solvable near $t=0$. It could be seen as a quasi-homogeneous version of Mizohata operator $D_t+it\vert D_x\vert$ since its symbol is $$ i\tau-t\vert \xi\vert^2=i(\tau+it\vert \xi\vert^2) $$ and at $\tau =0=t,\vert\xi\vert=1$, we have {$\tau,t\vert \xi\vert^2$}$=\vert \xi\vert^2>0$.

Answer 2

The following example is an second order differential operator with real coefficients defined on $\mathbb{R}^{3}$ which is not solvable about origin:

$Pu=(x_{2}^{2}-x_{3}^{2})D_{1}^{2}u+(1+x_{1}^{2})(D_{2}^{2}u-D_{3}^{2}u)-$ $$x_{1}x_{2}D_{1}D_{2}u-D_{1}D_{2}(x_{1}x_{2}u)+x_{1}x_{3}D_{1}D_{3}u+D_{1}D_{3}(x_{1}x_{3}u)$$

This beautiful operator was given by H$\ddot{o}$rmander in his Springer 1963 'Linear Partial Differential Operators' book.