We consider the van der Waals' free energy functional, with scaling parameter e, in the plane domain R+ x R+, with inhomogeneous Dirichlet boundary conditions. We impose the two stable phases on the horizontal boundaries R+ x {0} and R+ x {+infinity}, and free boundary conditions on {+infinity} x R+. Finally, the datum on {0} x R+ is chosen in such a way that the interface between the pure phases is pinned at some point (0, y). We show that there exists a critical scaling, y = y(epsilon), such that, as epsilon -> 0, the competing effects of repulsion from the boundary and penalization of gradients play a role in determining the optimal shape of the (properly rescaled) interface. This result is achieved by means of an asymptotic development of the free energy functional. As a consequence, such analysis is not restricted to minimizers but also encodes the asymptotic probability of fluctuations.
BOUNDARY EFFECTS IN THE GRADIENT THEORY OF PHASE TRANSITIONS / BERTINI MALGARINI, Lorenzo; Butta', Paolo; Garroni, Adriana. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - STAMPA. - 44:2(2012), pp. 926-945. [10.1137/110821469]
BOUNDARY EFFECTS IN THE GRADIENT THEORY OF PHASE TRANSITIONS
BERTINI MALGARINI, Lorenzo;BUTTA', Paolo;GARRONI, Adriana
2012
Abstract
We consider the van der Waals' free energy functional, with scaling parameter e, in the plane domain R+ x R+, with inhomogeneous Dirichlet boundary conditions. We impose the two stable phases on the horizontal boundaries R+ x {0} and R+ x {+infinity}, and free boundary conditions on {+infinity} x R+. Finally, the datum on {0} x R+ is chosen in such a way that the interface between the pure phases is pinned at some point (0, y). We show that there exists a critical scaling, y = y(epsilon), such that, as epsilon -> 0, the competing effects of repulsion from the boundary and penalization of gradients play a role in determining the optimal shape of the (properly rescaled) interface. This result is achieved by means of an asymptotic development of the free energy functional. As a consequence, such analysis is not restricted to minimizers but also encodes the asymptotic probability of fluctuations.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.