The two-well problem with surface energy

Abstract

Let Ω be a bounded Lipschitz domain in R 2 , let H be a 2 × 2 diagonal matrix with det( H ) = 1. Let ε > 0 and consider the functional over A F ∩ W 2,1 (Ω), where A F is the class of functions from Ω satisfying affine boundary condition F . It can be shown by convex integration that there exists F ∉ SO (2) ∪ SO (2) H and u ∈ A F with I 0 ( u ) = 0. Let 0 < ζ 1 < 1 < ζ 2 < ∞, . In this paper we begin the study of the asymptotics of m ε ≔ inf BF ∩ W 2,1 I ε for such F . This is one of the simplest minimization problems involving surface energy for which we can hope to see the effects of convex integration solutions. The only known lower bounds are lim inf ε→0 m ε /ε = ∞. We link the behaviour of m ε to the minimum of I 0 over a suitable class of piecewise affine functions. Let {τ i } be a triangulation of Ω by triangles of diameter less than h and let denote the class of continuous functions that are piecewise affine on a triangulation {τ i }. For the function u ∈ B F let be the interpolant, i.e. the function we obtain by defining ũ⌊τ i to be the affine interpolation of u on the corners of τ i . We show that if for some small ω > 0 there exists u ∈ B F ∩ W 2,1 with then, for h = ε (1+6399ω)/3201 , the interpolant satisfies I 0 (ũ) ≤ h 1− c ω . Note that it is trivial that , so we reduce the problem of non-trivial (scaling) lower bounds on m ε /ε to the problem of non-trivial lower bounds on .