By a geometric variational problem I mean a problem of minimising a functional defined on k-dimensional geometric objects, like currents or varifolds, lying in n-dimensional ambient space. I focus on functionals defined as integrals, where the integrand depends on the point and the tangent k-plane at that point. One example is the k-dimensional Hausdorff measure generated by some non-Euclidean norm on .

Ellipticity (AE), introduced by Almgren in the 1960s, is a condition on the functional ensuring existence and partial regularity of minimisers. The atomic condition (AC) was defined a few years ago by G. De Philippis, A. De Rosa, and F. Ghiraldin so to ensure rectifiability of critical points. Together with A. De Rosa we proved that (AC) implies (AE).

The problem with this theory is that there are virtually no specific non-trivial examples of functionals satisfying either (AE) or (AC). A. De Rosa and R. Tione proposed recently the scalar atomic condition (SAC) which might be easier to verify.

In my talk I shall review definitions, properties, and relations between conditions (AE), (AC), and (SAC). I shall talk about my joint work with A. De Rosa (CPAM 2020) and also about ongoing work with my student Mariusz Janosz.