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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.

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Boundary value problems in Lipschitz domains for equations with lower order coefficients
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by Georgios Sakellaris PDF
Trans. Amer. Math. Soc. 372 (2019), 5947-5989 Request permission

Abstract:

We use the method of layer potentials to study the $R_2$ regularity problem and the $D_2$ Dirichlet problem for second order elliptic equations with lower order coefficients in bounded Lipschitz domains. For $R_2$ we establish existence and uniqueness by assuming that $\mathcal {L}$ is of the form $\mathcal {L}u=-\text {div}(A\nabla u+bu)+c\nabla u+du$, where the matrix $A$ is uniformly elliptic and Hölder continuous, $b$ is Hölder continuous, and $c,d$ belong to Lebesgue classes and satisfy either the condition $d\geq \ \text {div} b$ or $d\geq \ \text {div} c$ in the sense of distributions. In particular, $A$ is not assumed to be symmetric, and there is no smallness assumption on the norms of the lower order coefficients. We also show existence and uniqueness for $D_2$ for the adjoint equations $\mathcal {L}^tu=0$.
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Additional Information
  • Georgios Sakellaris
  • Affiliation: Department of Mathematics, Universitat Autònoma de Barcelona, Bellaterra 08193, Barcelona, Spain
  • Email: gsakellaris@mat.uab.cat
  • Received by editor(s): September 18, 2018
  • Received by editor(s) in revised form: April 26, 2019
  • Published electronically: July 30, 2019
  • Additional Notes: The author received funding from the European Union’s Horizon 2020 research and innovation program under Marie Skłodowska-Curie grant agreement no. 665919, and he is partially supported by MTM-2016-77635-P (MICINN, Spain) and 2017 SGR 395 (Generalitat de Catalunya).
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 5947-5989
  • MSC (2010): Primary 35J25; Secondary 31B25
  • DOI: https://doi.org/10.1090/tran/7895
  • MathSciNet review: 4014299