MATHEMATICA BOHEMICA, Vol. 124, No. 2–3, pp. 193-219 (1999)

The obstacle problem for functions of least gradient

William P. Ziemer, Kevin Zumbrun

William P. Ziemer, Kevin Zumbrun, Department of Mathematics, Indiana University, Bloomington, IN 47405-5701, USA, e-mails:,

Abstract: For a given domain $\Omega\subset\Bbb{R}^n$, we consider the variational problem of minimizing the $L^1$-norm of the gradient on $\Omega$ of a function $u$ with prescribed continuous boundary values and satisfying a continuous lower obstacle condition $u\ge\psi$ inside $\Omega$. Under the assumption of strictly positive mean curvature of the boundary $\partial\Omega$, we show existence of a continuous solution, with Hölder exponent half of that of data and obstacle.
This generalizes previous results obtained for the unconstrained and double-obstacle problems. The main new feature in the present analysis is the need to extend various maximum principles from the case of two area-minimizing sets to the case of one sub- and one superminimizing set. This we accomplish subject to a weak regularity assumption on one of the sets, sufficient to carry out the analysis. Interesting open questions include the uniqueness of solutions and a complete analysis of the regularity properties of area superminimizing sets. We provide some preliminary results in the latter direction, namely a new monotonicity principle for superminimizing sets, and the existence of "foamy" superminimizers in two dimensions.

Keywords: least gradient, sets of finite perimeter, area-minimizing, obstacle

Classification (MSC2000): 49Q05, 35J85

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