Geodesic
The obstacle problem for functions of least gradient
Mathematica Bohemica, Tome 124 (1999) no. 2-3, pp. 193-219

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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 Holder 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.
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 Holder 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.
DOI : 10.21136/MB.1999.126244
Classification : 35J85, 35R35, 49Q05
Keywords: least gradient; sets of finite perimeter; area-minimizing sets; obstacle 
Ziemer, William P.; Zumbrun, Kevin. The obstacle problem for functions of least gradient. Mathematica Bohemica, Tome 124 (1999) no. 2-3, pp. 193-219. doi: 10.21136/MB.1999.126244
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