PreprintCC BY 4.0arXiv
Giaquinta's example revisited: the Liouville property for a class of nonuniformly elliptic equations in low dimensions
Abstract
We consider entire solutions u: ℝn → ℝ, n ≥ 2, of the equation div(∇ f(∇ u)) = 0 with convex density f: ℝn → ℝ given for example by f(∇ u) = ∑i=1n |∂i u|pi with exponents pi ≥ 2 including Giaquinta's example for the choice p1 = … = pn−1 = 2, pn = 4. If n ≤ 2 + min{pi} holds, then we have the Liouville property: the boundedness of u implies its constancy.
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BibTeX
@misc{bildhauer2026giaquintas,
title = {{Giaquinta's example revisited: the Liouville property for a class of nonuniformly elliptic equations in low dimensions}},
author = {Michael Bildhauer and Martin Fuchs},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11369},
eprint = {2610.11369},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11369},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Giaquinta's example revisited: the Liouville property for a class of nonuniformly elliptic equations in low dimensions
AU - Michael Bildhauer
AU - Martin Fuchs
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11369
UR - https://arxiv.org/abs/2610.11369
AB - We consider entire solutions u: ℝ^n → ℝ, n ≥ 2, of the equation div(∇ f(∇ u)) = 0 with convex density f: ℝ^n → ℝ given for example by f(∇ u) = ∑_i=1^n |∂_i u|^p_i with exponents p_i ≥ 2 including Giaquinta's example for the choice p_1 = … = p_n−1 = 2, p_n = 4. If n ≤ 2 + min{p_i} holds, then we have the Liouville property: the boundedness of u implies its constancy.
ER -