The Obstacle Problem for Functions of Least Gradient
| dc.creator | Ziemer, William P. | |
| dc.creator | Zumbrun, Kevin | |
| dc.date | 1998-11-07 | |
| dc.date.accessioned | 2026-07-07T05:26:45Z | |
| dc.date.available | 2026-07-07T05:26:45Z | |
| dc.description | For a given domain $Ω\subset \Bbb{R}^n$, 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 $u\ge Ψ$ inside $Ω$. Under the assumption of strictly positive mean curvature of the boundary $\partialΩ$, we show existence of a continuous solution, with Hölder 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. | |
| dc.description | 27 Pages | |
| dc.identifier | https://arxiv.org/abs/math/9811042 | |
| dc.identifier | http://arxiv.org/abs/math/9811042 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/77670 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Primary 49Q05; Secondary 35J85 | |
| dc.title | The Obstacle Problem for Functions of Least Gradient | |
| dc.type | text |