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We provide a strengthening of an elementary technique in geometric measure theory. Given an s -set S ⊂ R , in the language of tangent measures, this technique establishes the existence of tangent measures to the measure H bS on one side of an (n− 1) -plane. If S is purely unrectifiable or of dimension less than n − 1 , our strengthening consists of being… (More)
We provide a different approach to and prove a (partial) generalisation of a recent theorem on the structure of low energy solutions of the compatible two well problem in two dimensions [Lor05], [CoSc06]. More specifically we will show that a “quantitative” two well Liouville theorem holds for the set of matrices K = SO (2) ∪ SO (2) H where H = ( σ 0 0 σ−1… (More)
In this note we give sharp lower bounds for a non-convex functional when minimised over the space of functions that are piecewise affine on a triangular grid and satisfy an affine boundary condition in the second lamination convex hull of the wells of the functional.
Given a connected Lipschitz domain Ω we let Λ(Ω) be the subset of functions in W 2,2 (Ω) whose gradient (in the sense of trace) satisfies ∇u(x)·ηx = 1 where ηx is the inward pointing unit normal to ∂Ω at x. The functional Iǫ(u) = 1 2 R Ω ǫ −1 ˛ ˛ ˛1 − |∇u| 2 ˛ ˛ ˛ 2 + ǫ ˛ ˛ ∇ 2 u ˛ ˛ 2 minimised over Λ(Ω) serves as a model in connection with problems in… (More)