Lorentz p-harmonic functions are causal functions on Minkowski spacetime minimizing the energy $-\frac{1}{p}\int |du|^p_*$ with $p<1$, where $|du|_*$ is the Lorentzian norm of the differential. The operator associated with this energy is the p-d'Alembertian. Unlike the wave operator, it is elliptic, and it has recently played a key role in new proofs of the splitting theorem for Lorentzian spacetimes with nonnegative timelike Ricci curvature.

These functions behave very differently from their Euclidean counterparts. Causality is a one-sided constraint: causal functions are the Lorentzian analogue of monotone functions on the real line, to which they reduce in dimension $1+0$, and one should therefore expect them to jump. Since $p<1$, the energy does not charge jump discontinuities, so such jumps are not penalized. The natural functional setting is thus BV rather than Sobolev, and minimizers may be discontinuous.

In this talk I will present the main properties of Lorentz p-harmonic functions, focusing on two aspects. The first is the structure of the jump set: for $0<p<1$, jumps occur only across null hypersurfaces and are entirely determined by the boundary datum. The second is a set of sufficient conditions on the boundary datum, Lorentzian analogues of the classical bounded slope condition, which guarantee existence, uniqueness and $C^{1,\alpha}$ regularity of minimizers.

Based on joint work with G. De Philippis, N. Gigli, R. J. McCann, A. Ohanyan and M. Zanardini.