An Introduction to the Obstacle Problem
Abstract
We give a self-contained account of the classical obstacle problem, written for graduate students who wish to enter the field of free boundary problems. Starting from a one-dimensional model and an explicit radially symmetric example, we set up the variational problem, show that minimizers exist and are unique, and prove interior W2,p-regularity as well as the optimal C1,1-regularity. The second half of the notes is devoted to the free boundary: we prove that it is porous, introduce blow-up limits, and use the Alt-Caffarelli-Friedman monotonicity formula to show that every free boundary point is either regular or singular. These notes grew out of a mini-course at the Rio Summer School on PDEs in 2020, and they are meant to be read alongside, not instead of, the monograph by Petrosyan, Shahgholian, and Uraltseva.
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BibTeX
@misc{minne2026introduction,
title = {{An Introduction to the Obstacle Problem}},
author = {Andreas Minne},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37943},
eprint = {2610.37943},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37943},
}
RIS
TY - UNPB
TI - An Introduction to the Obstacle Problem
AU - Andreas Minne
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37943
UR - https://arxiv.org/abs/2610.37943
AB - We give a self-contained account of the classical obstacle problem, written for graduate students who wish to enter the field of free boundary problems. Starting from a one-dimensional model and an explicit radially symmetric example, we set up the variational problem, show that minimizers exist and are unique, and prove interior W^2,p-regularity as well as the optimal C^1,1-regularity. The second half of the notes is devoted to the free boundary: we prove that it is porous, introduce blow-up limits, and use the Alt-Caffarelli-Friedman monotonicity formula to show that every free boundary point is either regular or singular. These notes grew out of a mini-course at the Rio Summer School on PDEs in 2020, and they are meant to be read alongside, not instead of, the monograph by Petrosyan, Shahgholian, and Uraltseva.
ER -