Matching boundary jets without interior equality

For every integer $m\ge 0$, differentiation gives $\frac{d^m}{d{t}^m}{e}^{-1\mathrm{/}t}=P_m(1\mathrm{/}t)e^{-1\mathrm{/}t}$ on $t>0$ for a polynomial $P_m$, by induction on $m$. For every polynomial $P$, $P(s)e^{-s}\to 0$ as $s\to \mathrm{\infty }$; for each power $s^N$, this follows from $e^s\ge s^{N+1}\mathrm{/}(N+1)!$. Thus every right derivative tends to zero at $t=0$, so extension by zero to $t\le 0$ is smooth and flat there. Composition with $t=1-4\mathrm{\mid }x{\mathrm{\mid }}^2$ proves $\psi \in C_c^{\mathrm{\infty }}(B(0,1))$, with compact support in $\overline{B(0,1\mathrm{/}2)}$. It is nonzero at $x=0$, and $\mathrm{\parallel }\psi {\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$.

The bound $\mathrm{\mid }ϵ\mathrm{\mid }\mathrm{\parallel }\psi {\mathrm{\parallel }}_{\mathrm{\infty }}<1$ gives ${\gamma }_2(x)\ge 1-\mathrm{\mid }ϵ\mathrm{\mid }\mathrm{\parallel }\psi {\mathrm{\parallel }}_{\mathrm{\infty }}>0$, so it satisfies the positivity condition in [F1]. Since $\psi$ vanishes on a neighborhood of $\mathrm{\partial }\mathrm{\Omega }$, ${\gamma }_1$ and ${\gamma }_2$ agree on that neighborhood and all their boundary derivatives agree. At the origin, ${\gamma }_2(0)-{\gamma }_1(0)=ϵe^{-1}\mathrm{\ne }0$. Thus identical boundary jets do not force interior equality.