Non-solvability in the flat category of elliptic operators with real analytic coefficients

Abstract

Let $\Omega \subset {\mathbb{R}}^n,n\ge 2$, be an open set. For an elliptic differential operator $L$ on $\Omega$ with real analytic coefficients and a point $p\in \Omega$, we construct a smooth function $g$ with the following properties: $g$ is flat at $p$ and the equation $Lu=g$ has no smooth local solution $u$ that is flat at $p$.

arXiv commentv3: final version, to appear in Annales de la Faculté des Sciences de Toulouse under the title "A local obstruction for elliptic operators with real analytic coefficients on flat germs". 15 pages, 1 figure