Modica–Mortola theorem

Remarks

The lower bound comes from the inequality $a^2 + b^2 \ge 2ab$, which gives $\varepsilon \abs{\nabla u}^2 + W(u)/\varepsilon \ge 2 \sqrt{W(u)} \abs{\nabla u} = \abs{\nabla (\phi \circ u)}$ with $\phi(t) = 2 \int_0^t \sqrt{W(s)} \, ds$. The lower semicontinuity of the total variation then bounds the limit from below by the perimeter of $E$ times $\phi(1) - \phi(0) = c_W$. For the recovery sequence, one puts the optimal one-dimensional profile across $\partial E$. Here $q(s) = 1/(1 + e^{-s})$ solves $q' = \sqrt{W(q)}$. When $\partial E$ is smooth, $u_\varepsilon(x) = q(d(x)/\varepsilon)$ works, with $d$ the signed distance to $\partial E$. The same result holds for other continuous double-well potentials $W \ge 0$ that vanish exactly at two points and grow suitably at infinity. Then $c_W$ is the integral of $2\sqrt{W}$ between the two wells.

The theorem is often combined with a volume constraint $\int_\Omega u \, dx = m$. Then the fundamental theorem of Γ-convergence shows that minimizers of $F_\varepsilon$ converge, along subsequences, to characteristic functions of sets of least perimeter with the given volume.

Going deeper

For a student

This is the model example of Γ-convergence and the reason it became popular. It connects the diffuse-interface energy behind the Allen–Cahn equation with the sharp-interface perimeter functional of geometric measure theory. The constant $c_W$ is the energy per unit area of the optimal one-dimensional transition profile. The result is due to Luciano Modica and Stefano Mortola, and Modica used it to study phase transitions with a volume constraint.