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The harmonic map problem with mixed boundary conditions

Author: Panayotis Smyrnelis
Journal: Proc. Amer. Math. Soc. 143 (2015), 1299-1313
MSC (2010): Primary 58320; Secondary 35J50
Published electronically: November 12, 2014
MathSciNet review: 3293743
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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

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
Article copyright: © Copyright 2014 American Mathematical Society

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