A geometric approach to nonlocal 2-Hessian equations
Abstract
We study a nonlocal 2-Hessian equation, given by an infimum of linear deformations of the fractional Laplacian, that is, infA∈ A2Δs (u∘ A)(A−1x). We characterize the class of coefficient matrices A2, which determines the behavior of the operator, and provide a detailed geometric description of the possible degeneracies. Our main theorem shows that the nonlocal 2-Hessian equation remains uniformly elliptic for strictly positive right-hand sides, which leads to regularity estimates. The results hold under weaker hypotheses than those previously considered in the literature, and for the full range s∈(0,1). In particular, no convexity on the solutions is required. Our hypotheses can be interpreted as nonlocal counterparts of the local notions of semiconcavity and 2-convexity. Moreover, all the results are stable as s→1, recovering the local case. The geometric methods developed here are new, even in the local setting, and may be relevant to a broader class of nonlocal fully nonlinear equations and curvature-type problems.
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BibTeX
@misc{charro2026geometric,
title = {{A geometric approach to nonlocal 2-Hessian equations}},
author = {Fernando Charro and María del Mar González and María Soria-Carro},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10195},
eprint = {2609.10195},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10195},
}
RIS
TY - UNPB
TI - A geometric approach to nonlocal 2-Hessian equations
AU - Fernando Charro
AU - María del Mar González
AU - María Soria-Carro
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10195
UR - https://arxiv.org/abs/2609.10195
AB - We study a nonlocal 2-Hessian equation, given by an infimum of linear deformations of the fractional Laplacian, that is, inf_A∈ A_2Δ^s (u∘ A)(A^−1x). We characterize the class of coefficient matrices A_2, which determines the behavior of the operator, and provide a detailed geometric description of the possible degeneracies. Our main theorem shows that the nonlocal 2-Hessian equation remains uniformly elliptic for strictly positive right-hand sides, which leads to regularity estimates. The results hold under weaker hypotheses than those previously considered in the literature, and for the full range s∈(0,1). In particular, no convexity on the solutions is required. Our hypotheses can be interpreted as nonlocal counterparts of the local notions of semiconcavity and 2-convexity. Moreover, all the results are stable as s→1, recovering the local case. The geometric methods developed here are new, even in the local setting, and may be relevant to a broader class of nonlocal fully nonlinear equations and curvature-type problems.
ER -