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Transactions of the American Mathematical Society

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

 
 

 

Boundary value problems in Lipschitz domains for equations with lower order coefficients


Author: Georgios Sakellaris
Journal: Trans. Amer. Math. Soc. 372 (2019), 5947-5989
MSC (2010): Primary 35J25; Secondary 31B25
DOI: https://doi.org/10.1090/tran/7895
Published electronically: July 30, 2019
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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

DOI: https://doi.org/10.1090/tran/7895
Keywords: Boundary value problems, Lipschitz domain, nontangential maximal function, layer potentials, lower order coefficients
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).
Article copyright: © Copyright 2019 American Mathematical Society