## 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)

### Keywords:

second-order system; non-divergence form
Full Text:

### References:

  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  L. Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations 8 (1983), no. 1, 21-64. · Zbl 0546.35023  R. Regbaoui, Strong uniqueness for second order differential operators, J. Differential Equations 141 (1997), no. 2, 201-217. \inst{} · Zbl 0887.35040
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