The Obstacle Problem for Functions of Least Gradient
The Obstacle Problem for Functions of Least Gradient
复制标题
最小梯度函数的障碍问题
DOI：
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
K. Zumbrun
中科院分区：
文献类型：
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作者：
W. Ziemer;K. Zumbrun
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 $uge Psi$ inside $Omega$. Under the assumption of strictly positive mean curvature of the boundary $partialOmega$, we show existence of a continuous solution, with H"older 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.
