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

Abstract

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For a given domain Ω n , we consider the variational problem of minimizing the L 1 -norm of the gradient on Ω of a function u with prescribed continuous boundary values and satisfying a continuous lower obstacle condition u ψ inside Ω . Under the assumption of strictly positive mean curvature of the boundary Ω , 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.

How to cite

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Ziemer, 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 -

References

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