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On Poisson operators and Dirichlet-Neumann maps in $ H^s$ for divergence form elliptic operators with Lipschitz coefficients


Authors: Yasunori Maekawa and Hideyuki Miura
Journal: Trans. Amer. Math. Soc. 368 (2016), 6227-6252
MSC (2010): Primary 35J15, 35J25, 35S05
DOI: https://doi.org/10.1090/tran/6571
Published electronically: November 6, 2015
MathSciNet review: 3461032
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Abstract: We consider second order uniformly elliptic operators of divergence form in $ \mathbb{R}^{d+1}$ whose coefficients are independent of one variable. Under the Lipschitz condition on the coefficients we characterize the domain of the Poisson operators and the Dirichlet-Neumann maps in the Sobolev space $ H^s(\mathbb{R}^d)$ for each $ s\in [0,1]$. Moreover, we also show a factorization formula for the elliptic operator in terms of the Poisson operator.


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Additional Information

Yasunori Maekawa
Affiliation: Mathematical Institute, Tohoku University, 6-3 Aoba, Aramaki, Aoba, Sendai 980-8578, Japan
Email: maekawa@math.tohoku.ac.jp

Hideyuki Miura
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan
Address at time of publication: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Ookayama 2-12-1, Meguro-ku, Tokyo 152-8552, Japan
Email: miura@is.titech.ac.jp

DOI: https://doi.org/10.1090/tran/6571
Keywords: Divergence form elliptic operators, Poisson operators, Dirichlet-Neumann maps
Received by editor(s): September 17, 2013
Received by editor(s) in revised form: August 10, 2014
Published electronically: November 6, 2015
Article copyright: © Copyright 2015 American Mathematical Society