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Strong unique continuation property for some second order elliptic systems. (English) Zbl 1167.35321

From the introduction: Let \(\Omega\) be a nonempty open connected subset of \(\mathbb R^n\) containing the origin, and let
\[ P(x,\partial)= \sum_{1\leq j,k\leq n} A_{j,k}(x)\partial_j\partial_k \]
be an elliptic differential operator in \(\Omega\) where \(A_{j,k}\) is an \(N\times N\) matrix valued function with the entries which are Lipschitz continuous in \(\Omega\) for any \(1\leq j,k\leq n\). We assume that \(A_{j,k}^*A_{l,m}= A_{l,m}A_{j,k}^*\) for any \(1\leq j,k,l,m\leq n\), and there exist an elliptic differential operator \(p(\partial)= \sum_{1\leq j,k\leq n}a_{j,k}\partial_j\partial_k\) with real coefficients and complex numbers \(\lambda_j\), \(j=1,2,\dots,N\) such that
\[ P(0,\partial)= \text{diag} \begin{pmatrix} \lambda_1p(\partial) &&\\ &\ddots &\\ &&\lambda_Np(\partial) \end{pmatrix}. \]
Then it follows the following theorem:
Theorem. There exists a positive constant \(C^*\) depending only on \(p(\partial)\) such that if \(u\in\{H_{\text{loc}}^2(\Omega)\}^N\) satisfies
\[ |P(x,\partial)u|\leq C_0|u|/|x|^2+ C_1|\nabla u|/|x| \]
with \(C_1<C^*\) and
\[ \lim_{\rho\to0} \rho^{-\beta} \int_{|x|\leq\rho} |\partial_x^\alpha u|^2\,dx=0, \]
for any positive \(\beta\) and any \(|\alpha|\leq2\), then \(u\) is identically zero in some neighborhood of the origin.

MSC:

35B60 Continuation and prolongation of solutions to PDEs
35J45 Systems of elliptic equations, general (MSC2000)
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References:

[1] C. Grammatico, A result on strong unique continuation for the Laplace operator, Comm. Partial Differential Equations 22 (1997), no. 9-10, 1475-1491. · Zbl 0939.35049 · doi:10.1080/03605309708821308
[2] L. Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations 8 (1983), no. 1, 21-64. · Zbl 0546.35023 · doi:10.1080/03605308308820262
[3] R. Regbaoui, Strong uniqueness for second order differential operators, J. Differential Equations 141 (1997), no. 2, 201-217. \inst{} · Zbl 0887.35040 · doi:10.1006/jdeq.1997.3327
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