On the scaling of the two well problem
Andrew Lorent · 2005
Let H = ` σ 0 0 σ−1 for σ > 0. And let K := SO (2) ∪ SO (2)H. We establish a sharp relation between the following two minimisation problems. Firstly the two well problem with surface energy. Let q ≥ 1. Let I (u) = Z Ω d (Du (z) ,K) + Du (z) q dLz and let AF denote the subspace of functions in W 2,q (Ω) with det (Du (z)) ≥ 0 for a.e. z ∈ Ω and supz∈Ω ‖ [Du (z)]−1 ‖ ≤ C satisfying the affine boundary condition u (z) = F (z) for z ∈ ∂Ω, where F ∈ K. We consider the scaling (with respect to ) of m := inf u∈AF I (u) . Secondly the finite element approximation to the two well problem without surface energy. Let δ > 0 be any small number. Let Fh (u) = R Ω d “ Du (z) , N h δ 84 (K) ” dL2z. Let Bh F denote the space of functions that are piecewise affine on a triangular grid {τi} of grid size h satisfying the affine boundary condition u (z) = F (z) for z ∈ ∂Ω. We consider the scaling of αh := inf u∈Bh F Fh (u) . Let q ≥ 1. We will show that for any small h, for := hq we have αh ≥ ch 1 3 =⇒ m ≥ c′ 1 3q . Simple examples show αh ≤ Ch 1 3 and m ≤ C′ 1 3q so our theorem states that optimal (scaling) lower bounds on αh imply optimal (scaling) lower bounds on m q for any q ≥ 1. The main tool we will use to establish this reduction will be an Lq version of the suboptimal two well Liouville Theorem proved in [22]. We will give a simple proof of this result using the case of equality of the isoperimetric inequality. In addition for the case q = 1 we show optimal (scaling) lower bounds on I1 follow from optimal (scaling) lower bounds on F0 by applying the optimal two well Liouville Theorem of Conti, Schweizer [6].