Abstract
In this chapter, we study a classical counterexample due to Alinhac and Baouendi [5] for the wave equation with respect to a timelike hyperplane. We study as well an a priori unlikely consequence of the previous counterexample, namely Métivier’s counterexamples [105, 106], to Cauchy uniqueness for analytic non-linear systems. We show as well that a so-called compact uniqueness result can be achieved with a very weak hypothesis. This chapter ends with the formulation of various conjectures and open problems.
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Notes
- 1.
Here is the only instance in this section where coordinates on the cotangent bundle of \(\Omega \) are denoted by \((t, y,\tau , \eta )\). Later on the letter \(\tau \) is used as a large parameter.
- 2.
Indeed, we have for \(1\le l\le m-1\), \( \frac{4(l+1)}{l}=4+\frac{4}{l}\le 8 \).
- 3.
The concavity of \(x\mapsto 1-e^{-x}\) implies for \(x\in [0,2]\), \( 1-e^{-x}\ge \frac{1-e^{-2}}{2} x\ge \frac{x}{4}, \) yielding the first inequality, whereas the second inequality follows from the fact that the same function is increasing and thus for \(x\ge 1\), \(1-e^{-x}\ge 1-1/e> 1/2\) since \(e>2\).
- 4.
We have \(\nabla (e^{u})=e^{u}\nabla u,\) \(\nabla ^{2} (e^{u})=e^{u}\nabla ^{2} u+e^{u}\nabla u\otimes \nabla u,\) and checking Lemma B.8 for higher order derivatives, we find
$$ \nabla ^{m}(e^{u})= e^{u}\sum _{\begin{array}{c} 1\le r\le m\\ m_{1}+\cdots +m_{r}=m, m_{j}\ge 1 \end{array}} m!\prod _{1\le j\le r}\frac{\nabla ^{{m_{j}}}u}{m_{j}!} =e^{u}\nabla ^{m}u+e^{u}\Phi \bigl ((\nabla ^{l} u)_{1\le l\le m-1}\bigr ). $$ - 5.
Let \(\Omega \) be an open subset of \(\mathbb R^{n}\) and let W be a subset of \(\dot{T}^{*}(\Omega )=\Omega \times \bigl (\mathbb R^{n}\backslash \{0\}\bigr )\). We define \(S^{*}(\Omega )=\Omega \times \mathbb S^{n-1}\). The set W is said to be compact-conic if \(W\cap S^{*}(\Omega )\) is compact and if W is conic, i.e. \( (x,\xi )\in W, \lambda >0\Longrightarrow (x,\lambda \xi )\in W. \) In the case above, \(U_{0}\) is a compact neighborhood of \(x_{0}\) in \(\mathbb R^{n}\) and with \(\tilde{V}_{0}=V_{0}\cap \mathbb S^{n-1}\), we have \(V_{0}=\{\xi \in \mathbb R^{n}\backslash \{0\}, \xi /\vert \xi \vert \in \tilde{V}_{0}\}\).
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Lerner, N. (2019). On the Edge of Pseudo-convexity. In: Carleman Inequalities. Grundlehren der mathematischen Wissenschaften, vol 353. Springer, Cham. https://doi.org/10.1007/978-3-030-15993-1_6
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DOI: https://doi.org/10.1007/978-3-030-15993-1_6
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