Singular perturbation of a class of non-convex functionals

Authors:
Xinwei Yu;Zhiping Li;Lung-an Ying
Affiliations:
School of Mathematical Sciences, Peking University, People's Republic of China and Applied & Computational Mathematics, California Institute of Technology, Caltech 217-50, 91125 Pasadena, CA;School of Mathematical Sciences, Peking University, People's Republic of China;School of Mathematical Sciences, Peking University, People's Republic of China
Venue:
Nonlinear Analysis: Theory, Methods & Applications
Year:
2003

Citing 2
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Direct methods in the calculus of variations

Direct methods in the calculus of variations
Second order singular perturbation models for phase transitions

SIAM Journal on Mathematical Analysis

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Abstract

Models involving singular perturbation to a non-convex potential energy play a very important role in describing phase transitions, e.g. the celebrated Cahn-Hillard model where a two-well potential energy functional (i.e., the potential has two zeros) is perturbed by the L2-norm of the gradient.Many variants of this model have been studied. In this paper, we perturb a general multiwell energy functional by the L2-norm of a higher gradient Hessian of arbitrary order and study its Γ(L1)-limit. As expected, the limit functional assigns different surface energy densities to interfaces between different phases and computes the total energy.