Reflection, removable singularities, and approximation for partial differential equations. II

Abstract:

Let ${\Omega ^j}$ be domains in ${R^n}$. For each $j$ we are given a system ${{\mathbf {D}}^j}$ of linear constant coefficient operators and a function ${f^j}$ on ${\Omega ^j}$ satisfying ${{\mathbf {D}}^j}{f^j} = 0$. When the ${f^j}$ satisfy certain compatibility conditions on the intersections ${\Omega ^j} \cap {\Omega ^{j’}}$ then we can extend them so as to be solutions of ${{\mathbf {D}}^j}$ on larger domains. As a consequence of our methods we are able to sharpen Hartogs’ theorems to allow for continuation of solutions of overdetermined systems over noncompact sets.