𝐿^{𝑝} regularity of least gradient functions

Abstract

It is shown that in the anisotropic least gradient problem on an open bounded setΩ⊂RN\Omega \subset \mathbb {R}^Nwith Lipschitz boundary, given boundary dataf∈Lp(∂Ω)f \in L^p(\partial \Omega )the solutions lie inLNpN−1(Ω)L^{\frac {Np}{N-1}}(\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.