$\Gamma$-convergence, minimizing movements and generalized mean curvature evolution
Irene Fonseca, Markos A. Katsoulakis · Differential and Integral Equations · 1995
In this paper we show that viscosity solutions to curvature evolution equations may be obtained as limits of minimizers for -limits of inhomogeneous, anisotropic singular perturbations for certain nonconvex variational problems.We consider the energywhere ⌦ is an open, bounded, strongly Lipschitz domain of R N , u : ⌦ !R n , and W supports two phases; i.e., W has two isolated (global) minimum points a and b.The minimization of the energy E(•) subject to fixed volume fraction ✓, 0 < ✓ < 1, admits infinitely many solutions, which are piecewise constant measurable functions of the form u = A a + ( 1A )b, with meas(A) = ✓ meas(⌦).In order to find a selection criterion for resolving this nonuniqueness, we fix an initial phase-a configuration, A 0 , and we introduce the family of perturbed problems