Optimal regularity of the pressure field in incompressible optimal transport

Ascertain the optimal regularity of the pressure field p(t,x) appearing as the Lagrange multiplier in the multiphase formulation of incompressible optimal transport on the torus; in particular, prove or refute Brenier’s conjecture that the pressure p is semiconcave.

Background

In Brenier’s multiphase (incompressible) optimal transport formulation, the pressure field arises as a Lagrange multiplier enforcing incompressibility. Existence of a unique pressure as a distribution is known, and improved regularity (e.g., p∈L^2_t L^{d/(d−1)}_x) has been established.

Despite progress, the best possible regularity of p is unknown. Brenier conjectured that p is semiconcave, but this has not been proved, and current regularity is insufficient to justify certain formal second-variation computations for entropy along geodesics.