| Title:
|
The obstacle problem for functions of least gradient (English) |
| Author:
|
Ziemer, William P. |
| Author:
|
Zumbrun, Kevin |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
124 |
| Issue:
|
2 |
| Year:
|
1999 |
| Pages:
|
193-219 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
least gradient |
| Keyword:
|
sets of finite perimeter |
| Keyword:
|
area-minimizing sets |
| Keyword:
|
obstacle |
| MSC:
|
35J85 |
| MSC:
|
35R35 |
| MSC:
|
49Q05 |
| idZBL:
|
Zbl 0936.49024 |
| idMR:
|
MR1780692 |
| DOI:
|
10.21136/MB.1999.126244 |
| . |
| Date available:
|
2009-09-24T21:37:11Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126244 |
| . |
| Reference:
|
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