PreprintCC0arXiv
Lehto-type growth conditions and a Picard theorem for the gradient of entire solutions to the Euler -- Lagrange equations of variational integrals with linear growth in the plane including a small extension of Moser's Bernstein result
Abstract
We mainly consider entire solutions u: ℝ2 → ℝ of the Euler -- Lagrange equation for a variational integral with density g(|∇ u|) of linear growth. Assuming a Lehto-type condition at infinity, we show that either ∇ u is a constant vector or the image of ∇ u is the whole plane with the possible exception of at most one point. This result is extended to certain energy densities of nearly linear growth. Moreover, we slightly generalize the Bernstein result of Moser.
Related
- Uniqueness results for the exterior Dirichlet and Neumann problem in two dimensions for the Euler-Lagrange equations of variational integrals with linear growth
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- Blowup of smooth solutions to 2D quasilinear wave equations with higher-order null conditions and short pulse initial data
- Regularity for Minimizers of Mixed-Smoothness Variational Integrals
- The Liouville property for entire local minimizers from the space BVloc(R2,RN) of variational integrals with linear growth
BibTeX
@misc{fuchs2026lehtotype,
title = {{Lehto-type growth conditions and a Picard theorem for the gradient of entire solutions to the Euler -- Lagrange equations of variational integrals with linear growth in the plane including a small extension of Moser's Bernstein result}},
author = {Martin Fuchs},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01784},
eprint = {2611.01784},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01784},
note = {CC0},
}
RIS
TY - UNPB
TI - Lehto-type growth conditions and a Picard theorem for the gradient of entire solutions to the Euler -- Lagrange equations of variational integrals with linear growth in the plane including a small extension of Moser's Bernstein result
AU - Martin Fuchs
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01784
UR - https://arxiv.org/abs/2611.01784
AB - We mainly consider entire solutions u: ℝ^2 → ℝ of the Euler -- Lagrange equation for a variational integral with density g(|∇ u|) of linear growth. Assuming a Lehto-type condition at infinity, we show that either ∇ u is a constant vector or the image of ∇ u is the whole plane with the possible exception of at most one point. This result is extended to certain energy densities of nearly linear growth. Moreover, we slightly generalize the Bernstein result of Moser.
ER -