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Hessian operators, null Lagrangians and the integral comparison principle
Abstract
We investigate the conditions under which a Hessian operator satisfies the integral comparison principle. We establish a rather unexpected connection with the notion of a null Lagrangian from the calculus of variations. Under mild conditions we obtain a complete classification, showing that the only operators for which the integral comparison principle holds are linear combinations of k-Hessians and, in the homogeneous case, precisely the k-Hessians themselves. An analogous result is established for the complex Hessian operator, revealing in particular substantial obstructions to the development of a genuine pluripotential theory for the 𝒥-equation.
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BibTeX
@misc{dinew2026hessian,
title = {{Hessian operators, null Lagrangians and the integral comparison principle}},
author = {Sławomir Dinew and Szymon Myga and Marcin Sroka},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06313},
eprint = {2610.06313},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06313},
}
RIS
TY - UNPB
TI - Hessian operators, null Lagrangians and the integral comparison principle
AU - Sławomir Dinew
AU - Szymon Myga
AU - Marcin Sroka
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06313
UR - https://arxiv.org/abs/2610.06313
AB - We investigate the conditions under which a Hessian operator satisfies the integral comparison principle. We establish a rather unexpected connection with the notion of a null Lagrangian from the calculus of variations. Under mild conditions we obtain a complete classification, showing that the only operators for which the integral comparison principle holds are linear combinations of k-Hessians and, in the homogeneous case, precisely the k-Hessians themselves. An analogous result is established for the complex Hessian operator, revealing in particular substantial obstructions to the development of a genuine pluripotential theory for the 𝒥-equation.
ER -