Two-parameter Gamma convergence for the diffuse domain energies

Determine whether the two-parameter family of diffuse domain energy functionals \(\mathcal{E}_{\varepsilon,\alpha}\) Gamma-converges to the limiting energy \(\mathcal{E}_{0,0}\) as \((\varepsilon,\alpha)\to(0,0)\), and identify the appropriate topological space for this convergence.

Background

The paper establishes Gamma convergence of the diffuse-domain energies Eε,α\mathcal{E}_{\varepsilon,\alpha} to E0,α\mathcal{E}_{0,\alpha} only when α>0\alpha>0 is fixed and ε→0\varepsilon\to0. It separately resolves the singular limit E0,α→E0,0\mathcal{E}_{0,\alpha}\to\mathcal{E}_{0,0} as α→0\alpha\to0. The unresolved problem is to analyze the simultaneous two-parameter limit in which both regularization parameters vanish without prescribing a particular path.

The result would provide the variational foundation for rigorous convergence analysis of the fully practical Two-Parameter Diffuse Domain Method, in which diffuse-interface regularization and alpha regularization are refined together.

References

Is the following two-parameter Gamma convergence valid: ${_{\varepsilon,\alpha} \overset{\Gamma}{\longrightarrow} _{0,0}$, as $(\varepsilon,\alpha) \to (0,0)$? 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,” Q4; Figure 1 caption