Free boundary minimal surfaces and the Allen-Cahn equation
The Allen-Cahn equation is a semi-linear elliptic equation arising in the van der Waals-Cahn-Hilliard theory of phase transitions. Earlier fundamental work by De Giorgi, Modica, Sternberg etc. revealed intriguing relationship between the Allen-Cahn equation and the theory minimal surfaces. Based on the deep regularity theory by Hutchinson, Tonegawa and Wickramasekera, Guaraco recently introduced a new PDE approach to the existence of minimal surfaces via the Allen-Cahn equation and there have been substantial progress along this direction in the past few years. In this talk, we will consider the Allen-Cahn equation on bounded domains and describe some geometric and analytic aspects of the boundary behaviour of the limit-interfaces. This work is substantially supported by research grants from Hong Kong Research Grants Council and National Science Foundation China.