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An example in the gradient theory of phase transitions
Camillo De Lellis
We prove by giving an example that when $n\geq 3$ the asymptotic behavior of functionals $\int_\Omega \epsilon |\nabla^2 u|^2+(1-|\nabla u|^2)^2/\epsilon$ is quite different with respect to the planar case. In particular we show that the one-dimensional ansatz due to Aviles and Giga in the planar case is no longer true in higher dimensions.