Suppose that we have a $r$-order differential operator $L:C^r(B^n, \mathbb{R})\to C^0(B^n,\mathbb{R})$ where $B^n $ is the open unitary ball in $\mathbb{R}^n$ (we can assume for simplicity that it is elliptic). Let $A\subset B^n$ be open subset (in the simplest case we can assume it to be convex).
How many solutions has the equation $$Lf = 0$$ $$f|_{A} = g$$ where $g:A\to \mathbb{R}$, $L|_{A} g = 0$ ?.
In the case $L$ is the Cauchy-Riemann operator then solutions are holomorphic functions hence $g$ determines a unique $f$ defined over the whole ball. But I expect we can have wilder behaviours for different operators.
This question is motivated by a naive approach to PDEs on manifolds. Indeed one can in principle think of solving the PDE on one chart $A_1$ getting $g$, then on another chart $A_2$ one should consider some constraint of the above type, namely $f|_{A_1\cap A_2} = g|_{A_1\cap A_2}$, repeating this kind of computation for all the charts we would get a solution defined over the whole manifold.
