Boundary value problems in Lipschitz domains for equations with lower order coefficients
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- by Georgios Sakellaris PDF
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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 $\mathcal {L}^tu=0$.References
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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