# The obstacle problem for functions of least gradient

William P. Ziemer; Kevin Zumbrun

Mathematica Bohemica (1999)

- Volume: 124, Issue: 2-3, page 193-219
- ISSN: 0862-7959

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topZiemer, William P., and Zumbrun, Kevin. "The obstacle problem for functions of least gradient." Mathematica Bohemica 124.2-3 (1999): 193-219. <http://eudml.org/doc/248464>.

@article{Ziemer1999,

abstract = {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\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.},

author = {Ziemer, William P., Zumbrun, Kevin},

journal = {Mathematica Bohemica},

keywords = {least gradient; sets of finite perimeter; area-minimizing sets; obstacle; least gradient; sets of finite perimeter; area-minimizing sets; obstacle},

language = {eng},

number = {2-3},

pages = {193-219},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {The obstacle problem for functions of least gradient},

url = {http://eudml.org/doc/248464},

volume = {124},

year = {1999},

}

TY - JOUR

AU - Ziemer, William P.

AU - Zumbrun, Kevin

TI - The obstacle problem for functions of least gradient

JO - Mathematica Bohemica

PY - 1999

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 124

IS - 2-3

SP - 193

EP - 219

AB - 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\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.

LA - eng

KW - least gradient; sets of finite perimeter; area-minimizing sets; obstacle; least gradient; sets of finite perimeter; area-minimizing sets; obstacle

UR - http://eudml.org/doc/248464

ER -

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