Abstract
Let L be a second-order, homogeneous, constant (complex) coefficient elliptic system in \({\mathbb {R}}^n\). The goal of this article is to prove a Fatou-type result, regarding the a.e. existence of the nontangential boundary limits of any null-solution u of L in the upper half-space, whose nontangential maximal function satisfies an integrability condition with respect to the weighted Lebesgue measure (1 + |x′|n−1)−1 dx′ in \({\mathbb {R}}^{n-1}\equiv \partial {\mathbb {R}}^n_{+}\). This is the best result of its kind in the literature. In addition, we establish a naturally accompanying integral representation formula involving the Agmon-Douglis-Nirenberg Poisson kernel for the system L. Finally, we use this machinery to derive well-posedness results for the Dirichlet boundary value problem for L in \({\mathbb {R}}^n_{+}\) formulated in a manner which allows for the simultaneous treatment of a variety of function spaces.
Dedicated with great pleasure to Wolfgang Sprössig on the occasion of his 70th birthday for his many contributions to partial differential equations
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Acknowledgements
The first and second authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV2015-0554).
They also acknowledge support from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC agreement no. 615112 HAPDEGMT.
The third author has been supported in part by a Simons Foundation grant # 426669, the fourth author has been supported in part by Simons Foundation grants #318658 and #616050, while the fifth author has been supported in part by the Simons Foundation grant # 281566.
This work has been possible thanks to the support and hospitality of Temple University (USA), University of Missouri (USA), and ICMAT, Consejo Superior de Investigaciones Científicas (Spain). The authors express their gratitude to these institutions.
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Marín, J.J., Martell, J.M., Mitrea, D., Mitrea, I., Mitrea, M. (2019). A Fatou Theorem and Poisson’s Integral Representation Formula for Elliptic Systems in the Upper Half-Space. In: Bernstein, S. (eds) Topics in Clifford Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-23854-4_5
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DOI: https://doi.org/10.1007/978-3-030-23854-4_5
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