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

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- by Martin Dindoš, Sukjung Hwang and Marius Mitrea PDF
- Trans. Amer. Math. Soc.
**374**(2021), 3659-3701 Request permission

## 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
- ORCID: 0000-0002-6886-7677
- Email: M.Dindos@ed.ac.uk
**Sukjung Hwang**- Affiliation: Department of Mathematics, Yonsei University, Republic of Korea
- MR Author ID: 1144003
- Email: sukjung_hwang@yonsei.ac.kr
**Marius Mitrea**- Affiliation: Department of Mathematics, Baylor University, Waco, Texas
- MR Author ID: 341602
- ORCID: 0000-0002-5195-5953
- Email: Marius_Mitrea@baylor.edu
- 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
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 3659-3701 - MSC (2020): Primary 35J47, 35J57
- DOI: https://doi.org/10.1090/tran/8306
- MathSciNet review: 4237959