Timeline for Differential inequalities under which a flat function must be identically zero
Current License: CC BY-SA 4.0
8 events
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| May 13, 2021 at 14:05 | comment | added | Ali Taghavi | Thank you again for your interesting answer. For a vector field $X$ on a manifold $M$ , we consider the set $\mathcal{G}(X)$ consisting of all $p \in M$ such that $D(f)=X.f$ is a G.S operator at p(or with your terminology, $D$ satisfies Carleman estimate at $P$. What can be said about topology-dynamical properties of $\mathcal{G}(X)$ ?is it compact when $M$ is compact? Is it closed? is it flow invariant? | |
| May 9, 2021 at 20:36 | comment | added | Connor Mooney | Regarding unique continuation for non-elliptic operators, I am not sure- however, the following expository paper of Tataru may be a useful starting point: math.berkeley.edu/~tataru/papers/shortucp.ps | |
| May 9, 2021 at 20:33 | comment | added | Connor Mooney | The "strong unique continuation" property holds for operators of the form $Lu := \text{div}(A(x)\nabla u) + b(x)\cdot \nabla u + V(x)u$, provided $A$ is uniformly elliptic and Lipschitz, and $b,\,V$ are e.g. bounded. The Lipschitz regularity of $A$ cannot be dropped. See e.g. the work of Garofalo-Lin for an approach to this result based on a monotonicity formula (onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160400305) or the following paper of Koch and Tataru (math.berkeley.edu/~tataru/papers/esucp3.pdf) and the references therein for an approach based on Carleman estimates. | |
| May 9, 2021 at 16:11 | comment | added | Ali Taghavi | To be honnest, I was interested in this flat consideration(for derivational operator associated to a vector field since I was commenting on this MO answer mathoverflow.net/a/176509/36688 | |
| May 9, 2021 at 16:04 | comment | added | Ali Taghavi | interpretation for this set? | |
| May 9, 2021 at 16:04 | comment | added | Ali Taghavi | Thank you very much for your very interesting answer. Regarding the first part i think you consider $f:=f(-x)$. Regarding the second part do you think that every arbitrary elliptic operator on a manifold satisfy this vanishing property(aroundflat point)?I did not read the details of the linked paper you provided yet. As another question, let'us consider non elliptic operator associated to derivational operator defined by a vector field. We consider the set of all points$ p\in M$ whose small neighborhoods fulfill the Carleman estimate you mentioned. Do you think that there are some dynamic. | |
| May 9, 2021 at 1:36 | history | edited | Connor Mooney | CC BY-SA 4.0 |
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| May 9, 2021 at 1:27 | history | answered | Connor Mooney | CC BY-SA 4.0 |