I'm looking at minimizers of $$ E_{\epsilon}(u) = \int_{B_1} \frac{1}{2} |\nabla u|^2 + \frac{1}{2\epsilon^2} (1 - |u|^2)^2 $$ for $u: B_1 \to \mathbb{C}$ with $u(x) = x$ on $\partial B_1$. This is a Ginzburg-Landau functional. Let $\alpha_{\epsilon}$ denote the infinimum of $E_{\epsilon}(u)$ over all smooth functions with such a boundary condition.
Suppose that we're given $$ \alpha_{\epsilon} \leq -\pi \log(\epsilon) + C $$ for some $C$ fixed and independent of $\epsilon$, and such a value is achieved by $\{u_{\epsilon}\}$ for each $\epsilon > 0$. I want to show that $|u_{\epsilon}| \to 1$ a.e. in $B_1$ as $\epsilon \to 0$. By relatively straightforward bounds, I can show $L^1$ convergence and convergence in measure of the $\{|u_{\epsilon}|\}$'s, but this is strictly weaker than pointwise convergence a.e. (we can assume $|u_{\epsilon}| \leq 1$ everywhere by energy minimization). How would I get pointwise convergence? This feels like some geometric measure theory decomposition argument
See problem 6 in http://web.stanford.edu/~ochodosh/AllenCahnSummerSchool2019.pdf
for the full reference