Higher-order singular perturbation models for phase transitions
arxiv(2024)
摘要
Variational models of phase transitions take into account double-well
energies singularly perturbed by gradient terms, such as the Cahn-Hilliard free
energy. The derivation by Γ-convergence of a sharp-interface limit for
such energy is a classical result by Modica and Mortola. We consider a singular
perturbation of a double-well energy by derivatives of order k, and show that
we still can describe the limit as in the case k=1 with a suitable
interfacial energy density, in accord with the case k=1 and with the case
k=2 previously analyzed by Fonseca and Mantegazza. The main isssue is the
derivation of an optimal-profile problem on the real line describing the
interfacial energy density, which must be conveniently approximated by minimum
problems on finite intervals with homogeneous condition on the derivatives at
the endpoints up to order k-1. To that end a careful study must be carried on
of sets where sequences of functions with equibounded energy are “close to the
wells” and have “small derivatives”, in terms of interpolation inequalities
and energy estimates.
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