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Singular set estimates in quasilinear Alt-Caffarelli problems
Abstract
For a nonnegative, uniformly convex function Φ, we study nonnegative minimizers of the quasilinear Alt-Caffarelli energy ∫ΩΦ(∇ u)+χ{u>0}. Under mild regularity assumptions on Φ, we show that the codimension of the singular set of the free boundary is at least 2+η for a positive η. With a singular minimizer in three dimensions in a forthcoming work, this result is sharp as long as integer dimensions are considered. Our proof method is flexible, and we use it to also prove a similar result for minimizers of the p-Laplacian counterpart of the Alt-Caffarelli energy.
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BibTeX
@misc{rosa2026singular,
title = {{Singular set estimates in quasilinear Alt-Caffarelli problems}},
author = {Antonio De Rosa and Robin Neumayer and Hui Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34986},
eprint = {2610.34986},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34986},
}
RIS
TY - UNPB
TI - Singular set estimates in quasilinear Alt-Caffarelli problems
AU - Antonio De Rosa
AU - Robin Neumayer
AU - Hui Yu
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34986
UR - https://arxiv.org/abs/2610.34986
AB - For a nonnegative, uniformly convex function Φ, we study nonnegative minimizers of the quasilinear Alt-Caffarelli energy ∫_ΩΦ(∇ u)+χ_{u>0}. Under mild regularity assumptions on Φ, we show that the codimension of the singular set of the free boundary is at least 2+η for a positive η. With a singular minimizer in three dimensions in a forthcoming work, this result is sharp as long as integer dimensions are considered. Our proof method is flexible, and we use it to also prove a similar result for minimizers of the p-Laplacian counterpart of the Alt-Caffarelli energy.
ER -