Pathwise Gamma convergence with alpha equal to a power of epsilon

Determine whether the one-parameter family of diffuse domain energy functionals \(\mathcal{E}_{\varepsilon,\varepsilon^p}\) Gamma-converges to \(\mathcal{E}_{0,0}\) as \(\varepsilon\searrow0\) for every \(p\in(0,\infty)\), and identify the appropriate topological space for this convergence.

Background

A practically relevant refinement strategy links the two regularization parameters by a prescribed path, such as α=εp\alpha=\varepsilon^p. The paper leaves unresolved whether Gamma convergence holds along every such power-law path.

This question is a specialization of the broader simultaneous-limit problem and is important for determining whether particular parameter-selection rules used in computations converge to the correct limiting energy.

References

Suppose $p\in(0,\infty)$. Is the following one-parameter Gamma convergence valid: ${_{\varepsilon,\varepsilonp} \overset{\Gamma}{\longrightarrow} _{0,0}$, as $\varepsilon\searrow 0$? With respect to what topological space?

— The $α$-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem  (2609.01237 - Luong et al., 1 Sep 2026) in Section 2, subsection “Convergence Questions,” Q5