$L^p$ regularity of least gradient functions

Abstract:

It is shown that in the anisotropic least gradient problem on an open bounded set $\Omega \subset \mathbb {R}^N$ with Lipschitz boundary, given boundary data $f \in L^p(\partial \Omega )$ the solutions lie in $L^{\frac {Np}{N-1}}(\Omega )$; 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.