Abstract
In this chapter, we introduce the notion of strong unique continuation for solutions of elliptic equations. Under suitable assumptions on the potential V, a solution of an elliptic differential inequality of type \(\vert {\Delta u}\vert \le \vert {Vu}\vert \) which is flat at a given point should vanish on the connected component of that point. We start with results involving critical radial potentials such as \(V(x)=\vert x\vert ^{-2}\), using essentially some \(L^{2}\) Carleman estimates due to R. Regbaoui. [117–119]. To handle the critical case \(V\in L^{n/2}\), we give an exposition of the D. Jerison and C. Kenig result [68] involving \(L^{p}-L^{q}\) Carleman estimates with singular weights, following a method introduced by C. Sogge. in [139]. The last sections of this chapter are concerned with T. Wolff’s measure-theoretic lemma used to modify Carleman’s method and to obtain some weak unique continuation results with critical potentials.
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Notes
- 1.
The Lorentz space \(L^{p,{\infty }}\) is defined as the vector space of measurable functions V such that
$$ \sup _{\lambda>0} \lambda ^{p}\vert \{x, \vert V(x)\vert >\lambda \}\vert <+{\infty }, $$where \(\vert A\vert \) stands for the Lebesgue measure of A.
- 2.
Since the role of \(t', t''\) is symmetric in both formulas, we may assume \(t'\ge t''\) and the property is equivalent to the obvious equalities \(2t'-t'-t''=t'-t''=t'+t''-2t''.\)
- 3.
Note that when the potentials are radial, our Sect. 8.1 displays many strong unique continuation results with singular gradient potentials such as \(C/{\vert x\vert },\) with C small enough with respect to a dimensional constant.
- 4.
We have \(D_{\sigma ,\eta }=\cap _{\mathbf e\in \mathbb S^{n-1}}\bigl \{x\in \mathbb R^{n}, -\rho (\sigma , \eta , -\mathbf e)\le \langle x-g_{\eta },\mathbf e\rangle \le \rho (\sigma , \eta , \mathbf e)\bigr \}. \)
- 5.
Since \(\inf _{\mathbf e\in \mathbb S^{n-1}}\rho (\sigma , \eta , \mathbf e)=\rho _{1}>0,\) \(D_{\sigma , \eta }\supset \cap _{\mathbf e\in \mathbb S^{n-1}}\{x, \vert \langle x-g_{\eta },\mathbf e\rangle \vert \le \rho _{1}\}=\bar{B}(g_{\eta }, \rho _{1}).\)
- 6.
In fact, for each \(t\in \mathbb R\), there exists a \(\theta \in (0,1)\) such that
$$ e^{t}-1= t e^{\theta t}\Longrightarrow \vert e^{t}-1\vert \le \vert t\vert \max _{\theta \in [0,1]}e^{\theta t} =\vert t\vert \max (1, e^{t}). $$
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Lerner, N. (2019). Strong Unique Continuation Properties for Elliptic Operators. In: Carleman Inequalities. Grundlehren der mathematischen Wissenschaften, vol 353. Springer, Cham. https://doi.org/10.1007/978-3-030-15993-1_8
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DOI: https://doi.org/10.1007/978-3-030-15993-1_8
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