# Second Order Singular Perturbation Models for Phase Transitions

@article{Fonseca2000SecondOS,
title={Second Order Singular Perturbation Models for Phase Transitions},
author={Irene Fonseca and Carlo Mantegazza},
journal={SIAM J. Math. Anal.},
year={2000},
volume={31},
pages={1121-1143}
}
• Published 1 April 2000
• Mathematics
• SIAM J. Math. Anal.
Singular perturbation models involving a penalization of the first order derivatives have provided a new insight into the role played by surface energies in the study of phase transitions problems. It is known that if $W:{\mathbb R}^d \to [0,+\infty)$ grows at least linearly at infinity and it has exactly two potential wells of level zero at $a, b \in {\mathbb R}^d$, then the $\Gamma(L^1)$-limit of the family of functionals $${\mathcal F}_\varepsilon(u):= \begin{cases} \int_{\Omega} \left… Asymptotic analysis of a second-order singular perturbation model for phase transitions • Mathematics • 2009 We study the asymptotic behavior, as$${\varepsilon}$$tends to zero, of the functionals$${F^k_\varepsilon}$$introduced by Coleman and Mizel in the theory of nonlinear second-order materials; A Γ‐convergence result for the two‐gradient theory of phase transitions • Mathematics • 2002 The generalization to gradient vector fields of the classical double‐well, singularly perturbed functionals,$$ I_{\varepsilon} ( u;\Omega ) :=\int_{\Omega}{{1}\over{\varepsilon}} W(\nabla u)
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