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Approximation in norm and energy with bounded functions for autonomous variational problems
Abstract
We study the approximation of finite-energy Sobolev functions by bounded Sobolev functions for integral functionals associated with possibly signed Carathéodory integrands. The approximating sequence is required to converge strongly in the Sobolev space, while the corresponding energy densities converge in L1. Ordinary truncations may fail because they replace the gradient by zero on the tails, where the integrand need not be integrable. Our main contribution is a general method based on spatially dependent barriers whose gradients can be adapted to the structure of the Lagrangian.
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BibTeX
@misc{bousquet2026approximation,
title = {{Approximation in norm and energy with bounded functions for autonomous variational problems}},
author = {Pierre Bousquet and Carlo Mariconda and Giulia Treu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36869},
eprint = {2609.36869},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36869},
}
RIS
TY - UNPB
TI - Approximation in norm and energy with bounded functions for autonomous variational problems
AU - Pierre Bousquet
AU - Carlo Mariconda
AU - Giulia Treu
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36869
UR - https://arxiv.org/abs/2609.36869
AB - We study the approximation of finite-energy Sobolev functions by bounded Sobolev functions for integral functionals associated with possibly signed Carathéodory integrands. The approximating sequence is required to converge strongly in the Sobolev space, while the corresponding energy densities converge in L^1. Ordinary truncations may fail because they replace the gradient by zero on the tails, where the integrand need not be integrable. Our main contribution is a general method based on spatially dependent barriers whose gradients can be adapted to the structure of the Lagrangian.
ER -