Abstract
Given any strictly convex norm on
that is C1 in
we study the generalized Aviles-Giga functional
for
and
satisfying
Using, as in the euclidean case
the concept of entropies for the limit equation
we obtain the following. First, we prove compactness in Lp of sequences of bounded energy. Second, we prove rigidity of zero-energy states (limits of sequences of vanishing energy), generalizing and simplifying a result by Bochard and Pegon. Third, we obtain optimal regularity estimates for limits of sequences of bounded energy, in terms of their entropy productions. Fourth, in the case of a limit map in BV, we show that lower bound provided by entropy productions and upper bound provided by one-dimensional transition profiles are of the same order. The first two points are analogous to what is known in the euclidean case
and the last two points are sensitive to the anisotropy of the norm
Notes
1 Note in order to get the pointwise convergence in (Equation20(20)
(20) ) for every
we can not define
via the standard symmetric (across zero) kernel centered on θ0,
This is why we do the two step procedure of defining
then modifying it.