Conjectured Γ-limit for higher-order phase transitions with boundary line tension

Let $\Omega$ be the domain and let $u$ and $v$ denote the bulk and boundary phase variables, with $T u$ the trace of $u$ on $\partial\Omega$. Set

$$\mathcal{V}:=BV(\Omega;\{a,b\})\times BV(\partial\Omega;\{\alpha,\beta\}).$$

For $m$, $\sigma$, $L$, $V$, and $\zeta$ as in the source, define

$$c:=\inf\left\{\zeta\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{|f'(x)-f'(y)|^2}{|x-y|^2}\,dx\,dy+\int_{-\infty}^{\infty}V(f(x))\,dx:\ f\in H_{\mathrm{loc}}^{3/2}(\mathbb{R}),\ \lim_{x\to\infty}f(-x)=\alpha,\ \lim_{x\to\infty}f(x)=\beta\right\},$$

where

$$\zeta:=\inf\left\{\frac{\iint_{\mathbb{R}\times\mathbb{R}^{+}}|D^2u(x,y)|^2\,dx\,dy}{\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{|g'(x)-g'(y)|^2}{|x-y|^2}\,dx\,dy}:u\in H^2(\mathbb{R}\times\mathbb{R}^{+}),\ Tu(\cdot,0)=g\ \text{in }\mathbb{R}\right\}.$$

Higher-order line-tension conjecture. Under the hypotheses of Theorem, the sequence $\{\mathcal{F}_{\varepsilon}\}_{\varepsilon>0}$ $\Gamma$-converges as $\varepsilon\to0^+$ to

$$\mathcal{F}_0(u,v):=\begin{cases} \displaystyle m\operatorname{Per}_{\Omega}(E_a)+\sum_{z=a,b}\sum_{\xi=\alpha,\beta}\sigma(z,\xi)\mathcal{H}^{N-1}(\{Tu=z\}\cap\{v=\xi\})+cL\operatorname{Per}_{\partial\Omega}(F_\alpha),& (u,v)\in\mathcal{V},\\ \infty,&\text{otherwise.} \end{cases}$$

Moreover, $\zeta$ is independent of $g\in H_{\mathrm{loc}}^{3/2}(\mathbb{R})$ satisfying $\lim_{x\to\infty}g(-x)=\alpha$ and $\lim_{x\to\infty}g(x)=\beta$. The conjecture identifies the missing boundary-transition energy as a line-tension term proportional to the perimeter of the boundary phase interface; the surrounding discussion explains that the bulk and boundary contributions are known, while this nonlocal higher-order contribution is expected but not established.