Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


The $L^p$ Dirichlet boundary problem for second order elliptic Systems with rough coefficients
HTML articles powered by AMS MathViewer

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


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 $2$.

Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2020): 35J47, 35J57
  • Retrieve articles in all journals with MSC (2020): 35J47, 35J57
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:
  • Sukjung Hwang
  • Affiliation: Department of Mathematics, Yonsei University, Republic of Korea
  • MR Author ID: 1144003
  • Email:
  • Marius Mitrea
  • Affiliation: Department of Mathematics, Baylor University, Waco, Texas
  • MR Author ID: 341602
  • ORCID: 0000-0002-5195-5953
  • Email:
  • 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:
  • MathSciNet review: 4237959