• Allen-Cahn
• curves of maximal slope
• gamma-convergence
• mean curvature flow
• Modica-Mortola

In the thesis we introduce a notion of mean curvature motion based on the theory of curves of maximal slope (that was introduced by De Giorgi, Marino and Tosques to study gradient flows in metric spaces) in the toy model of radial symmetry.
Then we study the approximation of the mean curvature motion based on the Allen-Cahn equation, that is the rescaled gradient flow of the Modica-Mortola functionals. In this context, under the assumption of radial symmetry, we prove an estimate on the Gamma-liminf of the metric slopes of these functionals, that was proved by Roeger and Schaetzle without the radiality assumption when the space dimension is 2 or 3.
Finally, we point out some of the main problems that one should solve in order to prove the convergence of the curves of maximal slope of the Modica-Mortola functionals to the notion of mean curvature motion as a curve of maximal slope, in the radial case.