T11 — Sharp-Interface Γ-Convergence

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1. #1

   Let be the smoothness-to-double-well ratio. As , the rescaled boundary-morphology energy

   $$\mathcal{E}^\varepsilon_{\mathrm{bd}}(u) = \varepsilon \sum_{x,y} \mathbf{N}_t(x,y)(u(x) - u(y))^2 + \frac{1}{\varepsilon} \sum_x W(u(x))$$
2. #2

   Γ-converges to a perimeter functional

   $$\mathcal{F}_0(\chi_E) = c_W \cdot \mathrm{Per}_G(E)$$