Smooth compactly supported solutions of some underdetermined elliptic PDE, with gluing applications

TL;DR

This paper establishes conditions under which smooth, compactly supported solutions to certain underdetermined elliptic PDEs can be constructed and glued together, contrasting with traditional unique continuation results.

Contribution

It introduces a method to glue solutions of underdetermined elliptic PDEs to create new solutions with prescribed support properties.

Findings

Constructed smooth compactly supported solutions in the kernel of elliptic operators.

Demonstrated solutions can be glued to form new solutions matching original outside a region.

Provided applications to divergence free fields and TT-tensors in geometry and physics.

Abstract

We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators $P$ can be glued in a chosen region in order to obtain a new smooth solution. This new solution is exactly equal to the starting elements outside the gluing region. This result completely contrasts with the usual unique continuation for determined or overdetermined elliptic operators. As a corollary we obtain compactly supported solutions in the kernel of $P$ and also solutions vanishing in a chosen relatively compact open region. We apply the result for natural geometric and physics contexts such as divergence free fields or TT-tensors.