Isolated Singularities of k-Hessian Equations
Abstract
We classify isolated singularities of nonpositive, k-admissible classical solutions to k-Hessian equations without radial symmetry assumptions. For 1≤ k<n/2, we consider Fk[u]=(−u)p with 1<p≤ nk/(n−2k). Below the Serrin exponent, we obtain fundamental-solution asymptotics and characterize removability by the vanishing of the leading coefficient. At the Serrin exponent, we determine the exact logarithmic correction and its universal coefficient for every nonremovable singularity. In the critical dimension k=n/2, we establish logarithmic asymptotics and a removability criterion for Fk[u]=f(x,u,∇ u) with continuous sources satisfying 0≤ f(x,s,ξ)≤ C|ξ|ℓ e−αs, where α>0 and 0≤ℓ<n. The proofs combine Hessian-measure compactness, Wolff-potential estimates, a global Bôcher theorem, and a bounded-variation analysis of spherical extrema.
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BibTeX
@misc{ma2026isolated,
title = {{Isolated Singularities of k-Hessian Equations}},
author = {Shiguang Ma and Shengyang Zang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29079},
eprint = {2610.29079},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29079},
}
RIS
TY - UNPB
TI - Isolated Singularities of k-Hessian Equations
AU - Shiguang Ma
AU - Shengyang Zang
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29079
UR - https://arxiv.org/abs/2610.29079
AB - We classify isolated singularities of nonpositive, k-admissible classical solutions to k-Hessian equations without radial symmetry assumptions. For 1≤ k<n/2, we consider F_k[u]=(−u)^p with 1<p≤ nk/(n−2k). Below the Serrin exponent, we obtain fundamental-solution asymptotics and characterize removability by the vanishing of the leading coefficient. At the Serrin exponent, we determine the exact logarithmic correction and its universal coefficient for every nonremovable singularity. In the critical dimension k=n/2, we establish logarithmic asymptotics and a removability criterion for F_k[u]=f(x,u,∇ u) with continuous sources satisfying 0≤ f(x,s,ξ)≤ C|ξ|^ℓ e^−αs, where α>0 and 0≤ℓ<n. The proofs combine Hessian-measure compactness, Wolff-potential estimates, a global Bôcher theorem, and a bounded-variation analysis of spherical extrema.
ER -