Critical points of functionalized Lagrangians

Keith Promislow; Hang Zhang

doi:10.3934/dcds.2013.33.1231

Article Contents

2013, Volume 33, Issue 4: 1231-1246. Doi: 10.3934/dcds.2013.33.1231

This issue Previous Article Next Article Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups

Critical points of functionalized Lagrangians

Keith Promislow^1, and
Hang Zhang^2,

1.
Department of Mathematics, Michigan State University, East Lansing, MI, 48824
2.
Nico Holding LLC, 222 W. Adams Street, Chicago, IL, 60606

Received: August 31, 2011

Revised: September 30, 2011

Published: March 2013

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Abstract

We present a novel class of higher order energies motivated by the study of network formation in binary mixtures of functionalized polymers and solvent. For a broad class of Lagrangians, we introduce their functionalized form, which is a higher order energy balancing the square of the variational derivative against the original energy. We show that the functionalized energies have global minimizers over several natural spaces of admissible functions. The critical points of the functionalized Lagrangian contain those of the original Lagrangian, however we demonstrate that for a sufficient strength of the functionalization all the critical points of the original Lagrangian are saddle points of the functionalized Lagrangian, and the global minima is a new structure.

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Mathematics Subject Classification: Primary: 49J45, 35Q74, 35Q56.

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