A variational Approach for Quasi-Periodic Fronts in Allen-Cahn Model Equations
In contrast to periodic functions, the class of quasi-periodic functions usually does not provide a good framework for formulating a variational problem. Difficulties arise from lack of suitable compactness, the problem of small divisors and so on.
In this lecture I first present an example in which a direct variational method can successfully capture a solution of quasi-periodic nature. More precisely, I consider a stationary Allen-Cahn model equation on the plane having spatially periodic coefficient:
-Δ u+f (x,y,u) = 0, (x,y) ∈ ℝ2
and discuss the existence of a near-planar front solution whose angle has irrational tangent. The existence of such a front can be derived from results of Moser (1986), Bangert (1989) and others. Their approach is first to prove the existence for rational angles (which can be done by direct minimization arguments in the class of periodic functions), and then to use an approximation argument to pass to the limit. However, no variational approach has been given to find such an irrational front directly. In a recent joint work with Paul Rabinowitz, we have succeeded in finding a direct variational approach to capture such solutions.
I will then discuss another varitational problem associated with traveling waves in spatially quasi-periodic media, namely the mini-max characterization of the speed of traveling waves. For simplicity I consider a one-dimensional Allen-Cahn equation:
ut = uxx+f (x,u) .
Such mini-max characterization has been known in the case of spatially periodic medium (Heinze-Papanicolaou-Stevens, 2001), in which case the solution of the mini-max problem coincides with the travelling wave. However, in the case of quasi-periodic medium, this mini-max problem may not have a solution at all. Intriguingly, the mini-max characterization still provides correct information about the speed of traveling waves despite lack of solutions.