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Gradient estimates for semilinear elliptic systems and other related results

Published online by Cambridge University Press:  22 October 2015

Panayotis Smyrnelis*
Affiliation:
Department of Mathematics, University of Athens, Panepistemiopolis, 15784 Athens, Greece ([email protected])

Abstract

A periodic connection is constructed for a double well potential defined in the plane. This solution violates Modica's estimate as well as the corresponding Liouville theorem for general phase transition potentials. Gradient estimates are also established for several kinds of elliptic systems. They allow us to prove the Liouville theorem in some particular cases. Finally, we give an alternative form of the stress–energy tensor for solutions defined in planar domains. As an application, we deduce a (strong) monotonicity formula.

MSC classification

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2015 

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