A result of existence and uniqueness for the Allen-Cahn equation with singular potentials and dynamic boundary conditions

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• Luca Calatroni, (Cambridge Centre for Analysis)
• Wednesday 14 November 2012, 16:00-17:00
• MR14, Centre for Mathematical Sciences.

If you have a question about this talk, please contact Kevin Crooks.

In this talk we will present well-posedness results for the solution to an initial and boundary-value problem for an Allen-Cahn type equation describing the phenomenon of phase transition for a material contained in a bounded and regular domain. The nonlinearity appearing in the equation is frequently assumed to be a double-well potential. In the talk we will generalize such nonlinearity considering a possibly non-smooth potential with domain different from the whole real line. Such potentials are typically called singular potentials. Even though the Allen-Cahn equation is frequently coupled with homogeneous Neumann boundary conditions (which are meant to represent the orthogonality of the interface to the boundary and the absence of mass flux), physicists have recently introduced the so-called dynamic boundary conditions which take into account the kinetics of the process on the boundary as well. In this talk we will consider similar boundary conditions where another singular potential appears. Both the uniqueness and the continuous dependence on data results hold in a quite general setting, whereas for the proof of the existence we need to enforce our assumptions, introducing a compatibility condition between the two singular potentials. We prove the existence result by regularizing the nonlinearities using Yosida approximations and exploiting some a priori estimates that allow us to pass to the limit thanks to compactness and monotonicity results.

This is a joint work with Prof. Pierluigi Colli, University of Pavia.

This talk is part of the Cambridge Analysts' Knowledge Exchange series.

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