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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The $ L^p$ Dirichlet boundary problem for second order elliptic Systems with rough coefficients


Authors: Martin Dindoš, Sukjung Hwang and Marius Mitrea
Journal: Trans. Amer. Math. Soc. 374 (2021), 3659-3701
MSC (2020): Primary 35J47, 35J57
DOI: https://doi.org/10.1090/tran/8306
Published electronically: February 2, 2021
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Abstract: Given a domain above a Lipschitz graph, we establish solvability results for strongly elliptic second-order systems in divergence-form, allowed to have lower-order (drift) terms, with $ L^p$-boundary data for $ p$ near $ 2$ (more precisely, in an interval of the form $ \big (2-\varepsilon ,\frac {2(n-1)}{n-2}+\varepsilon \big )$ for some small $ \varepsilon >0$). The main novel aspect of our result is that the coefficients of the operator do not have to be constant, or have very high regularity; instead they will satisfy a natural Carleson condition that has appeared first in the scalar case. A significant example of a system to which our result may be applied is the Lamé system for isotropic inhomogeneous materials. We show that our result applies to isotropic materials with Poisson ratio $ \nu <0.396$.

Dealing with genuine systems gives rise to substantial new challenges, absent in the scalar case. Among other things, there is no maximum principle for general elliptic systems, and the De Giorgi-Nash-Moser theory may also not apply. We are, nonetheless, successful in establishing estimates for the square-function and the nontangential maximal operator for the solutions of the elliptic system described earlier, and use these as alternative tools for proving $ L^p$ solvability results for $ p$ near $ 2$.


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Additional Information

Martin Dindoš
Affiliation: School of Mathematics, The University of Edinburgh and Maxwell Institute of Mathematical Sciences, United Kingdom
Email: M.Dindos@ed.ac.uk

Sukjung Hwang
Affiliation: Department of Mathematics, Yonsei University, Republic of Korea
Email: sukjung_hwang@yonsei.ac.kr

Marius Mitrea
Affiliation: Department of Mathematics, Baylor University, Waco, Texas
Email: Marius_Mitrea@baylor.edu

DOI: https://doi.org/10.1090/tran/8306
Keywords: Strongly elliptic system, boundary value problems, Carleson condition
Received by editor(s): September 15, 2018
Received by editor(s) in revised form: January 13, 2020, June 25, 2020, and September 21, 2020
Published electronically: February 2, 2021
Article copyright: © Copyright 2021 American Mathematical Society