The $\alpha$-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem

Abstract

We study the singular limit of a family of linear degenerate interface transmission problems arising from a regularization procedure in the newly proposed Two-Parameter Diffuse Domain Method (DDM2p). For $\alpha>0$, the regularized problem admits a strictly convex variational formulation on $H^{1}(\Omega)$. In the limit $\alpha\to0$, the problem degenerates to a weakly coupled interface system with a nonstandard energy structure. To characterize the limit, we introduce a closed Hilbert subspace $\mathcal{H}\subset H^{1}(\Omega)$, defined through an auxiliary Helmholtz problem on an annular subdomain $\Omega_2\subset \Omega$, and identify the limiting energy functional $\mathcal{E}_{0}$ on $\mathcal{H}$. We prove that the regularized energies $\mathcal{E}_{\alpha}$ $\Gamma$-converge to $\mathcal{E}_{0}$ in the strong $L^{2}(\Omega)$ topology, using the standard framework. Consequently, minimizers of $\mathcal{E}_{\alpha}$ converge to the unique minimizer of $\mathcal{E}_{0}$, which is shown to be equivalent to the solution of the limiting interface problem. We further prove strong convergence $u_{\alpha}\to u_{0}$ in $H^{1}(\Omega)$ and establish an $O(\alpha)$ convergence rate. Numerical experiments in one spatial dimension confirm the predicted first-order convergence rate and suggest that this rate is sharp.