Suppose $\Omega$ is a open bounded Lipschitz domain in $\mathbb{R}^n.$ Consider the sequences of energies

$$E_\varepsilon (u) := \int_\Omega \varepsilon |\nabla u|^2 + \frac{1}{\varepsilon} (1 - u^2)^2 \,dx.$$ It is then well known that if $u_\varepsilon$ is a sequence of minimizers of $E_\varepsilon$ subject to say a volume constraint, then $u_\varepsilon$ converge in $L^1$ to some $u_* \in BV(\Omega;\{\pm 1\}),$ and moreover one has the lower bound $$ \liminf_{\varepsilon \to 0} E_\varepsilon(u_\varepsilon) \geq c_0 \mathcal{H}^{n-1} (\partial\{u_* = 1\} \cap \Omega) $$ for an appropriate constant $c_0 > 0.$ My question is, is there a quantitative version of this inequality. For instance, is it known/true that for all sufficiently small $\varepsilon$ one has

$$ E_\varepsilon(u_\varepsilon) \geq c_0 \mathcal{H}^{n-1} (\partial\{u_* = 1\}) + o(\varepsilon)? $$

References would be greatly appreciated.

Suppose $\Omega\subset\mathbb{R}^n$ is an open, bounded Lipschitz domain. Consider the sequences of energies

$$E_\varepsilon (u) := \int_\Omega \varepsilon |\nabla u|^2 + \frac{1}{\varepsilon} (1 - u^2)^2 \,dx.$$ It is well-known that if $u_\varepsilon$ is a sequence of minimizers of $E_\varepsilon$ subject to say a volume constraint, then $u_\varepsilon$ converge in $L^1$ to some $u_* \in BV(\Omega;\{\pm 1\})$. Moreover, one has the lower bound $$ \liminf_{\varepsilon \to 0} E_\varepsilon(u_\varepsilon) \geq c_0 \mathcal{H}^{n-1} (\partial\{u_* = 1\} \cap \Omega) $$ for an appropriate constant $c_0 > 0$.

QUESTION. Is there a quantitative version of this inequality. For instance, is it known/true that for all sufficiently small $\varepsilon$ one has $$ E_\varepsilon(u_\varepsilon) \geq c_0 \mathcal{H}^{n-1} (\partial\{u_* = 1\}) + o(\varepsilon)? $$

References would be greatly appreciated.

Suppose $\Omega$ is a open bounded Lipschitz domain in $\mathbb{R}^n.$ Consider the sequences of energies

$$E_\varepsilon (u) := \int_\Omega \varepsilon |\nabla u|^2 + \frac{1}{\varepsilon} (1 - u^2)^2 \,dx.$$ It is then well known that if $u_\varepsilon$ is a sequence of minimizers of $E_\varepsilon$ subject to say a volume constraint, then $u_\varepsilon$ converge in $L^1$ to some $u_* \in BV(\Omega;\{\pm 1\}),$ and moreover one has the lower bound $$ \liminf_{\varepsilon \to 0} E_\varepsilon(u_\varepsilon) \geq c_0 \mathcal{H}^{n-1} (\partial\{u_* = 1\}) $$ for an appropriate constant $c_0 > 0.$ My question is, is there a quantitative version of this inequality. For instance, is it known/true that for all sufficiently small $\varepsilon$ one has

$$ E_\varepsilon(u_\varepsilon) \geq c_0 \mathcal{H}^{n-1} (\partial\{u_* = 1\}) + o(\varepsilon)? $$

References would be greatly appreciated.

Suppose $\Omega$ is a open bounded Lipschitz domain in $\mathbb{R}^n.$ Consider the sequences of energies

$$E_\varepsilon (u) := \int_\Omega \varepsilon |\nabla u|^2 + \frac{1}{\varepsilon} (1 - u^2)^2 \,dx.$$ It is then well known that if $u_\varepsilon$ is a sequence of minimizers of $E_\varepsilon$ subject to say a volume constraint, then $u_\varepsilon$ converge in $L^1$ to some $u_* \in BV(\Omega;\{\pm 1\}),$ and moreover one has the lower bound $$ \liminf_{\varepsilon \to 0} E_\varepsilon(u_\varepsilon) \geq c_0 \mathcal{H}^{n-1} (\partial\{u_* = 1\} \cap \Omega) $$ for an appropriate constant $c_0 > 0.$ My question is, is there a quantitative version of this inequality. For instance, is it known/true that for all sufficiently small $\varepsilon$ one has

$$ E_\varepsilon(u_\varepsilon) \geq c_0 \mathcal{H}^{n-1} (\partial\{u_* = 1\}) + o(\varepsilon)? $$

References would be greatly appreciated.