Sharp-interface limits beyond the regular-interface class

Extend the simultaneous sharp-interface limit for the diffuse-interface Landau–de Gennes droplet model to finite-perimeter droplets and tensor fields in the space $SBV^2$, including a characterization of the effective density associated with possible internal jumps.

Background

The paper establishes matching lower and upper bounds for the simultaneous limit of the phase-field thickness and exterior $Q$-tensor relaxation length within a regular-interface class. Specifically, the limiting droplet is assumed to have a $C^2$ boundary and the limiting tensor field belongs to $H^1(A;\mathcal S_0)$.

The authors explicitly identify extending the result to finite-perimeter droplets and $SBV^2$ tensor fields as unresolved. Such an extension would need to account for the effective interfacial contribution generated by internal jump sets of the tensor field, which is not treated by the present regular-interface theory.