PreprintCC BY 4.0arXiv
A note on existence of smooth solution to Jacobian equation for compactly supported smooth data in ℝ2
Abstract
In this note, we give an explicit construction of global solutions to the prescribed Jacobian equation det(∇ u)=f in ℝ2, for a class of data. For every f∈ Cc1(ℝ2) and p>1, we construct a solution u∈ Ẇ1,2p(ℝ2)∩ L∞(ℝ2). In particular, no sign condition or integral constraint is imposed on the compactly supported data. For f∈ Cc∞(ℝ2), the construction yields a smooth solution. We also consider rapidly decaying data and prove the existence of global solutions with bounded gradient for every f∈𝒮(ℝ2). Finally, we extend the construction to compactly supported data that are measurable in one variable and C1 in the other. Our proof relies on a similar idea as in [Moser, Trans. Amer. Math. Soc. 1965].
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BibTeX
@misc{jana2026note,
title = {{A note on existence of smooth solution to Jacobian equation for compactly supported smooth data in $\mathbbR^2$}},
author = {Animesh Jana},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14407},
eprint = {2609.14407},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14407},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A note on existence of smooth solution to Jacobian equation for compactly supported smooth data in ℝ^2
AU - Animesh Jana
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.14407
UR - https://arxiv.org/abs/2609.14407
AB - In this note, we give an explicit construction of global solutions to the prescribed Jacobian equation det(∇ u)=f in ℝ^2, for a class of data. For every f∈ C_c^1(ℝ^2) and p>1, we construct a solution u∈ Ẇ^1,2p(ℝ^2)∩ L^∞(ℝ^2). In particular, no sign condition or integral constraint is imposed on the compactly supported data. For f∈ C_c^∞(ℝ^2), the construction yields a smooth solution. We also consider rapidly decaying data and prove the existence of global solutions with bounded gradient for every f∈𝒮(ℝ^2). Finally, we extend the construction to compactly supported data that are measurable in one variable and C^1 in the other. Our proof relies on a similar idea as in [Moser, Trans. Amer. Math. Soc. 1965].
ER -