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Borderline double phase functionals with powers of the logarithm: regularity in the vectorial case
Abstract
We study vector-valued local minimizers of the borderline double phase functional whose integrand is |Dw|p+a(x)|Dw|plogγ(e+|Dw|), where p>1, γ>0 and the coefficient a(·) is nonnegative and bounded. If the modulus of continuity ω(·) of a(·) satisfies ω(r)logγ(1/r)→0 as r→0, then minimizers are locally Hölder continuous for every exponent less than one. If a(·) is Hölder continuous, then the gradients of minimizers are locally Hölder continuous. In the vector-valued setting these results were previously available only for γ=1. The proof combines a comparison with frozen functionals, whose regularity theory applies uniformly in the frozen coefficient, with the higher integrability of minimizers.
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BibTeX
@misc{choi2026borderline,
title = {{Borderline double phase functionals with powers of the logarithm: regularity in the vectorial case}},
author = {Youlin Choi and Jehan Oh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16900},
eprint = {2610.16900},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16900},
}
RIS
TY - UNPB
TI - Borderline double phase functionals with powers of the logarithm: regularity in the vectorial case
AU - Youlin Choi
AU - Jehan Oh
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16900
UR - https://arxiv.org/abs/2610.16900
AB - We study vector-valued local minimizers of the borderline double phase functional whose integrand is |Dw|^p+a(x)|Dw|^plog^γ(e+|Dw|), where p>1, γ>0 and the coefficient a(·) is nonnegative and bounded. If the modulus of continuity ω(·) of a(·) satisfies ω(r)log^γ(1/r)→0 as r→0, then minimizers are locally Hölder continuous for every exponent less than one. If a(·) is Hölder continuous, then the gradients of minimizers are locally Hölder continuous. In the vector-valued setting these results were previously available only for γ=1. The proof combines a comparison with frozen functionals, whose regularity theory applies uniformly in the frozen coefficient, with the higher integrability of minimizers.
ER -