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The obstacle problem for functions of least gradient. (English) Zbl 0936.49024
Summary: For a given domain $$\Omega \subset \mathbb R^n$$, 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 $$u\geq \psi$$ 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.

##### MSC:
 49Q05 Minimal surfaces and optimization 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
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