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Liouville-type theorem for the Lamé system with singular coefficients


Authors: Blair Davey, Ching-Lung Lin and Jenn-Nan Wang
Journal: Proc. Amer. Math. Soc. 147 (2019), 2619-2624
MSC (2010): Primary 35B60, 35B53; Secondary 74B05
DOI: https://doi.org/10.1090/proc/14409
Published electronically: March 1, 2019
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Abstract: In this paper, we study a Liouville-type theorem for the Lamé system with rough coefficients in the plane. Let $ u$ be a real-valued two-vector in $ \mathbb{R}^2$ satisfying $ \nabla u\in L^p(\mathbb{R}^2)$ for some $ p>2$ and the equation $ \operatorname {div}\left (\mu \left [\nabla u+(\nabla u)^T\right ]\right ) \allowbreak +\nabla (\lambda \operatorname {div} u)=0$ in $ \mathbb{R}^2$. When $ \Vert\nabla \mu \Vert _{L^2(\mathbb{R}^2)}$ is not large, we show that $ u\equiv$$ \text {constant}$ in $ \mathbb{R}^2$. As by-products, we prove the weak unique continuation property and the uniqueness of the Cauchy problem for the Lamé system with small $ \Vert\mu \Vert _{W^{1,2}}$.


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Additional Information

Blair Davey
Affiliation: Department of Mathematics, City College of New York (CUNY), New York, New York 10031
Email: bdavey@ccny.cuny.edu

Ching-Lung Lin
Affiliation: Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan
Email: cllin2@mail.ncku.edu.tw

Jenn-Nan Wang
Affiliation: Institute of Applied Mathematical Sciences, NCTS, National Taiwan University, Taipei 106, Taiwan
Email: jnwang@math.ntu.edu.tw

DOI: https://doi.org/10.1090/proc/14409
Received by editor(s): June 4, 2018
Received by editor(s) in revised form: September 27, 2018
Published electronically: March 1, 2019
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2019 American Mathematical Society