Calculus of Variations and Geometric Measure Theory

R. Monti - F. Serra Cassano

Degenerate perturbations of a two-phase transition model

created on 20 Dec 2002
modified on 29 Nov 2003


Published Paper

Inserted: 20 dec 2002
Last Updated: 29 nov 2003

Journal: J. Convex Anal.
Volume: 10
Number: 1
Pages: 1-34
Year: 2003


We study the $\Gamma-$convergence as $\e\to0^+$ of the family of degenerate functionals \[ Q_\e(u)=\e\int_\O\langle ADu,Du\rangle\, dx +\frac{1}{\e}\int_\O W(u)\,dx \] where $A(x)$ is a symmetric, \it non negative \rm $n\times n$ matrix on $\O$ (i.e. $\langle A(x)\xi,\xi\rangle\geq0$ for all $x\in\O$ and $\xi\in\Rn$) with regular entries and $W:\R\to[0,+\infty)$ is a double well potential having two isolated minimum points. Moreover, under suitable assumptions on the matrix $A$, we obtain a minimal interface criterion for the $\Ga-$limit functional exploiting some tools of Analysis in Carnot-Carathéodory spaces. We extend some previous results obtained for the non degenerate perturbations $Q_\e$ in the classical gradient theory of phase transiti

Keywords: phase transitions, $\Gamma$-convergence, Carnot-Carathéodory spaces