A mathematical foundation for QUMOND

Abstract

We link the QUMOND theory with the Helmholtz-Weyl decomposition and introduce a new formula for the gradient of the Mondian potential using singular integral operators. This approach allows us to demonstrate that, under very general assumptions on the mass distribution, the Mondian potential is well-defined, once weakly differentiable, with its gradient given through the Helmholtz-Weyl decomposition. Furthermore, we establish that the gradient of the Mondian potential is an $L^p$ vector field. These findings lay the foundation for a rigorous mathematical analysis of various issues within the realm of QUMOND. Given that the Mondian potential satisfies a second-order partial differential equation, the question arises whether it has second-order derivatives. We affirmatively answer this question in the situation of spherical symmetry, although our investigation reveals that the regularity of the second derivatives is weaker than anticipated. We doubt that a similarly general regularity result can be proven without symmetry assumptions. In conclusion, we explore the implications of our results for numerous problems within the domain of QUMOND, thereby underlining their potential significance and applicability.