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Take any region omega and let function u defined inside omega be the
distance from the boundary, u solves the iconal equation \lt|Du\rt|=1 with
boundary condition zero. Functional u is also conjectured (in some cases
proved) to be the "limiting minimiser" of various functionals that
arise models of blistering and micro magnetics. The precise formulation of
these problems involves the notion of gamma convergence. The Aviles Giga
functional is a natural "second order" generalisation of the Cahn
Hilliard model which was one of the early success of the theory of gamma
convergence. These problems turn out to be surprisingly rich with connections
to a number of areas of pdes. We will survey some of the more elementary
results, describe in detail of one main problems in field and state some
partial results.