Plateau Angle Conditions for the Vector-Valued Allen--Cahn Equation

Abstract

Under proper hypotheses, we rigorously derive the Plateau angle conditions at triple junctions of diffused interfaces in three dimensions, starting from the vector-valued Allen-Cahn equation with a triple-well potential. Our derivation is based on an application of the divergence theorem using the divergence-free form of the equation via an associated stress tensor.

Some basic facts on the system $\Delta u - W_{u} (u) = 0$

“…The stress-energy tensor T was introduced in joint unpublished work with Betelú [2] (see also the M.Sc. thesis of Dimitroula [11] for further elaboration) where it is utilized for deriving the Plateau angle conditions and was motivated by the work of Bronsard and Reitich [8].…”

“…In the context of the Ginzburg-Landau system, we note the book of Sandier and Serfaty [25], where this tensor has been utilized in a number of ways. Our source is Alikakos and Betelú [2] where the tensor was introduced specifically for writing the system as a divergence-free condition in order to derive the Plateau angle conditions.…”

On the structure of phase transition maps for three or more coexisting phases

“…For the phase transition case and for a triple-well potential, Gui [36] has derived such a result in the planar case n = 2, m = 2, thus relating the angles of a triple junction to the surface energies. This was extended to the three-dimensional case n = 3, m = 3, in [5]. Related also is the work of Kowalczyk, Liu, and Pacard [39].…”

Ποιοτική Μελέτη Λύσεων Μη Γραμμικών Ελλειπτικών Συστημάτων

“…Anal. (όρα [4]). Η επέκταση του αποτελέσματος στις τέσσερεις διαστάσεις καθιστά εμφανή την ύπαρξη ενός επαγωγικού βήματος για την εξαγωγή του στη γενική περίπτωση των n διαστάσεων με μοναδικό, ίσως, εμπόδιο τις δυσχέρειες που επιφέρει ο, ούτως ή άλλως, βαρύς και απαιτητικός συμβολισμός.…”

“…Επιπλέον, οι τρεις διεπιφάνειες συντρέχουν κατά μήκος ενός συνόλου συνδιάστασης 2 (ενός επιπέδου στην περίπτωσή μας) το οποίο καλούμε ράχη της τριπλής συμβολής. μελετήθηκε στο επίπεδο στα [19] και [30], ενώ πρόσφατα μελετήθηκε και στις τρεις διαστάσεις (όρα [4]).…”