A minimization problem for a cooperative system with a degenerate Bernoulli weight
Abstract
In this paper, we study local minimizers of the energy functionalJ(u) = ∫D (|∇ u|2 + Q2(x)χΩu) dx which give rise to a singular cooperative system. Here u=(u1,…,um): D→R+m is a vector-valued unknown function, Ωu:={|u|>0} is the positive set, χΩu is the characteristic function of the positive set Ωu, and Q(x) is the Bernoulli weight function. This problem was first introduced by Caffarelli, Shahgholian, and Yeressian in the poineer work ({\it Duke Math. J.} \textbf{167}(10), 2018), where the regularity theory for minimizers and for the free boundary ∂Ωu was established for nondegenerate Bernoulli weights, i.e., Q(x)≥ Qmin>0. The present paper studies the same free boundary problem but for a degenerate Bernoulli weight. The main difficulty lies in the coupling between the degeneracy of Q(x) and the vector-valued nature of the problem. We show that the the free boundary set ∂Ωu∩{Q(x)=0} decomposes into a nondegenerate part and a degenerate part. Nondegenerate points admit nontrivial blow-up limits; degenerate points have zero weighted density. We classify nondegenerate points into single-phase, multi-phase non-branching, and multi-phase branching points, analyzing their geometric structure. The branching-point singular structure for multi-phase points is purely vectorial and new in this setting. Finally, following Naber--Valtorta (Ann. Math. 185, 2017) and Edelen--Engelstein (Trans. Amer. Math. Soc. 371, 2019), we prove the nondegenerate set has locally finite ℋn−2-measure and is countably (n−2)-rectifiable.
Related
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BibTeX
@misc{du2026minimization,
title = {{A minimization problem for a cooperative system with a degenerate Bernoulli weight}},
author = {Lili Du and Chunlei Yang and Jing Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.25544},
eprint = {2609.25544},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25544},
}
RIS
TY - UNPB
TI - A minimization problem for a cooperative system with a degenerate Bernoulli weight
AU - Lili Du
AU - Chunlei Yang
AU - Jing Yang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2609.25544
UR - https://arxiv.org/abs/2609.25544
AB - In this paper, we study local minimizers of the energy functionalJ(u) = ∫_D (|∇ u|^2 + Q^2(x)χ_Ω_u) dx which give rise to a singular cooperative system. Here u=(u_1,…,u_m): D→R_+^m is a vector-valued unknown function, Ω_u:={|u|>0} is the positive set, χ_Ω_u is the characteristic function of the positive set Ω_u, and Q(x) is the Bernoulli weight function. This problem was first introduced by Caffarelli, Shahgholian, and Yeressian in the poineer work ({\it Duke Math. J.} \textbf{167}(10), 2018), where the regularity theory for minimizers and for the free boundary ∂Ω_u was established for nondegenerate Bernoulli weights, i.e., Q(x)≥ Q_min>0. The present paper studies the same free boundary problem but for a degenerate Bernoulli weight. The main difficulty lies in the coupling between the degeneracy of Q(x) and the vector-valued nature of the problem. We show that the the free boundary set ∂Ω_u∩{Q(x)=0} decomposes into a nondegenerate part and a degenerate part. Nondegenerate points admit nontrivial blow-up limits; degenerate points have zero weighted density. We classify nondegenerate points into single-phase, multi-phase non-branching, and multi-phase branching points, analyzing their geometric structure. The branching-point singular structure for multi-phase points is purely vectorial and new in this setting. Finally, following Naber--Valtorta (Ann. Math. 185, 2017) and Edelen--Engelstein (Trans. Amer. Math. Soc. 371, 2019), we prove the nondegenerate set has locally finite ℋ^n−2-measure and is countably (n−2)-rectifiable.
ER -