$L^p$ regularity of least gradient functions

It is shown that solutions to the anisotropic least gradient problem for boundary data $f\in L^p(\mathrm{\partial }\mathrm{\Omega })$ lie in $L^{\frac{Np}{N-1}}(\mathrm{\Omega })$; the exponent is shown to be optimal. Moreover, the solutions are shown to be locally bounded with explicit bounds on the rate of blow-up of the solution near the boundary in two settings: in the anisotropic case on the plane and in the isotropic case in any dimension.