Interfaces separating two phases are thin rather than perfectly sharp. Allen–Cahn-type equations model interfaces as narrow transition layers, while minimal surfaces and mean curvature flow arise as their sharp-interface limits. This thesis develops a regularity theory for diffuse interfaces without relying on variational assumptions associated with the energy. Exact equations connecting the discrepancy with the curvature of level surfaces yield epsilon-regularity and diffuse Schauder estimates in both elliptic and parabolic settings, including both the classical and free-boundary variants.

The talk is open to members of the university.