The theory of partial regularity for elliptic systems replaces the classical De Giorgi–Nash–Moser theory for scalar equations, asserting that solutions are regular outside a negligible closed subset, called the singular set. In some cases, Hausdorff dimension estimates for this set can also be obtained. In general, the singular set is non-empty.

This theory is classical, originating in the work of Giusti–Miranda and Morrey, and in turn relying on De Giorgi’s seminal ideas for minimal surfaces. I shall present a few results aimed at extending the classical local partial regularity theory to nonlinear integrodifferential systems, and at providing some basic general tools for proving so-called epsilon-regularity theorems in nonlocal settings.

Based on recent joint work with Cristiana De Filippis (Parma) and Simon Nowak (Bielefeld); see also our survey paper in Discrete and Continuous Dynamical Systems 57 (2026).

Building on recent advances in nonlocal hyperelasticity, we discuss a class of variational problems involving integral functionals with nonlocal gradients. Specific to our set-up is a space-dependent interaction range that vanishes at the boundary of the reference domain. This ensures that the operator depends only on values within the domain and localizes to the classical gradient at the boundary, which allows for a seamless integration of nonlocal modeling with local boundary values. Our main contribution is a comprehensive study of the associated Sobolev spaces, including embedding results, the analysis of a trace operator and the proof of a Poincaré inequality. A central aspect of our technical approach lies in exploiting connections with pseudodifferential operator theory. As an application, we establish the existence of minimizers for functionals with quasiconvex or polyconvex integrands depending on heterogeneous nonlocal gradients, subject to local Dirichlet-, Neumann- or mixed-type boundary conditions.

The classical Stefan problem aims to describe the evolution of the moving boundary between two phases of a material undergoing a phase change, for instance the melting of ice to water. One way to study the problem is the so-called enthalpy formulation. Weak solutions (temperature) can be found in proper function spaces. My talk concerns the first key property beyond the original functional analytic set-up, namely the continuity of weak solutions. I will present some recent advances in this direction.