Timeline for A question on the proof of unique continuation for the case $u\in H^{2}$ in Le Rousseau, Lebeau and Robbiano book on Carleman estimates

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Jun 3 at 4:08	comment	added	monotone operator		Thanks!@WillieWong, I get it. Your answers and comments are invaluable for me!
Jun 3 at 4:03	comment	added	Willie Wong		Yes. The analogy with bootstrap arguments is apt. In fact, it may be easier to think in terms of the "closed" case where the extension is $u = 0$ on $\overline{\mathscr{B}_{t}}$ implies $u = 0$ on $\mathscr{B}_{t+\epsilon}$. The difference exposed is then that $u = 0$ is a closed condition, and when $u$ is by assumption continuous, you have that $u$ vanishing on the open ball implies it vanishes on the closed ball. This inability to take closures is fundamentally what breaks when you look at the ODE case.
Jun 3 at 4:02	vote	accept	monotone operator
Jun 3 at 4:00	comment	added	monotone operator		thank you for your answer, according to your explanation, Lebeau's argument is like the bootstrap in spirit, right? Proposition5.1 tell us that if $u$ is equal to $0$ on one side of hypersurface then $u$ will also equal to $0$ on a small neighborhood of that hypersurface($u$ satisfy the above partial differential inequalities). Your explanation about the relation between the proof of Lebeau and extension of solution of ode is really excellent!
Jun 3 at 3:49	history	answered	Willie Wong	CC BY-SA 4.0