Mathematics – Analysis of PDEs
Scientific paper
1998-11-07
Mathematics
Analysis of PDEs
27 Pages
Scientific paper
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 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.
Ziemer William P.
Zumbrun Kevin
No associations
LandOfFree
The Obstacle Problem for Functions of Least Gradient does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The Obstacle Problem for Functions of Least Gradient, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Obstacle Problem for Functions of Least Gradient will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-560809