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St. Petersburg Mathematical Journal

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Fatou-type theorems and boundary value problems for elliptic systems in the upper half-space


Authors: J. M. Martell, D. Mitrea, I. Mitrea and M. Mitrea
Original publication: Algebra i Analiz, tom 31 (2019), nomer 2.
Journal: St. Petersburg Math. J. 31 (2020), 189-222
MSC (2010): Primary 31A20, 35C15, 35J57, 42B37, 46E30; Secondary 35B65, 42B25, 42B30, 42B35
DOI: https://doi.org/10.1090/spmj/1592
Published electronically: February 4, 2020
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Abstract: This is a survey of recent progress in a program which to date has produced several publications and is aimed at proving general Fatou-type results and establishing the well-posedness of a variety of boundary value problems in the upper half-space  $ {\mathbb{R}}^n_+$ for second-order, homogeneous, constant complex coefficient, elliptic systems $ L$, formulated in a manner that emphasizes pointwise nontangential boundary traces of the null-solutions of $ L$ in $ {\mathbb{R}}^n_+$.


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Additional Information

J. M. Martell
Affiliation: Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13–15, E-28049 Madrid, Spain
Email: chema.martell@icmat.es

D. Mitrea
Affiliation: Baylor University, Department of Mathematics, One Bear Place #97328, Waco Texas 76798
Email: Dorina_Mitrea@baylor.edu

I. Mitrea
Affiliation: Department of Mathematics, Temple University, 1805NorthBroadStreet, Philadelphia, Pennsylvania 19122
Email: imitrea@temple.edu

M. Mitrea
Affiliation: Baylor University, Department of Mathematics, One Bear Place #97328, Waco Texas 76798
Email: Marius_Mitrea@baylor.edu

DOI: https://doi.org/10.1090/spmj/1592
Keywords: Fatou-type theorem, Dirichlet boundary value problem, elliptic system, Poisson kernel, nontangential maximal operator, nontangential boundary trace, Muckenhoupt weights, Hardy space, bounded mean oscillation, vanishing mean oscillation, subcritical growth, sublinear growth
Received by editor(s): November 25, 2018
Published electronically: February 4, 2020
Additional Notes: The first author acknowledges that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC agreement no. 615112 HAPDEGMT. He also acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554).
The second author was partially supported by Simons Foundation grant $#$426669
The third author was partially supported by Simons Foundation grants $#$318658 and #616050 and by the NSF grant #1900938
The fourth author was partially supported by Simons Foundation grant $#$637481
Dedicated: Dedicated with great pleasure and respect to Vladimir Maz’ya on the occasion of his $80$th birthday
Article copyright: © Copyright 2020 American Mathematical Society