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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The harmonic map problem with mixed boundary conditions
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by Panayotis Smyrnelis PDF
Proc. Amer. Math. Soc. 143 (2015), 1299-1313 Request permission

Abstract:

Given two polygons $S \subset \mathbb {R}^2$ and $\Sigma \subset \mathbb {R}^m$ with the same number of sides, we prove the existence and uniqueness of a smooth harmonic map $u:S \to \mathbb {R}^m$ satisfying the mixed boundary conditions for $S$ and $\Sigma$. This solution is constructed and characterized as a minimizer of the Dirichlet’s energy in the class of maps which satisfy the first mixed boundary condition. Several properties of the solution are established. We also discuss the mixed boundary conditions for harmonic maps defined in smooth domains of the plane.
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Additional Information
  • Panayotis Smyrnelis
  • Affiliation: Department of Mathematics, University of Athens, 11584 Athens, Greece
  • Email: ysmyrnelis@yahoo.fr
  • Received by editor(s): September 17, 2011
  • Received by editor(s) in revised form: August 6, 2012, and June 1, 2013
  • Published electronically: November 12, 2014
  • Additional Notes: The author was partially supported through the project PDEGE (Partial Differential Equations Motivated by Geometric Evolution), co-financed by the European Union European Social Fund (ESF) and national resources, in the framework of the program Aristeia of the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF)
  • Communicated by: James E. Colliander
  • © Copyright 2014 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 1299-1313
  • MSC (2010): Primary 58320; Secondary 35J50
  • DOI: https://doi.org/10.1090/S0002-9939-2014-12264-7
  • MathSciNet review: 3293743