Domain dependence of Sobolev-curl constants and structure of critical curl-curl solutions
Abstract
This paper investigates intrinsic critical Sobolev-curl constants on bounded domains and the structure of finite-energy solutions to the critical curl-curl equation in R^3. First, for 1< p < 3, we prove that the intrinsic critical p-curl constant on (0, a)\times D is bounded above by a^{p/3} times a Sobolev quotient on D. The proof uses an aligned test field whose optimal curl-free correction vanishes. Consequently, these constants tend to zero along thin convex boxes. For p = 2, we construct explicit convex and nonconvex domains with intrinsic constant below the sharp scalar Sobolev constant. The nonconvex example gives an affirmative answer to the domain-existence question in Problem 5 of Mederski and Szulkin(Arch. Ration. Mech. Anal. 241 (2021), 1815-1842), while the convex example disproves the lower bound stated there for convex domains. Concerning their Problem 2, we obtain an implicit Newtonian representation of finite-energy solutions and exclude two special forms of solutions. These structural results do not yet provide an explicit ground state or an unconditional ground-state decay rate as asked in Problem 2.
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BibTeX
@misc{liu2026domain,
title = {{Domain dependence of Sobolev-curl constants and structure of critical curl-curl solutions}},
author = {Yong Liu and Jun Wang and Xuan Wang and Wen Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14529},
eprint = {2610.14529},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14529},
}
RIS
TY - UNPB
TI - Domain dependence of Sobolev-curl constants and structure of critical curl-curl solutions
AU - Yong Liu
AU - Jun Wang
AU - Xuan Wang
AU - Wen Yang
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14529
UR - https://arxiv.org/abs/2610.14529
AB - This paper investigates intrinsic critical Sobolev-curl constants on bounded domains and the structure of finite-energy solutions to the critical curl-curl equation in R^3. First, for 1< p < 3, we prove that the intrinsic critical p-curl constant on (0, a)\times D is bounded above by a^{p/3} times a Sobolev quotient on D. The proof uses an aligned test field whose optimal curl-free correction vanishes. Consequently, these constants tend to zero along thin convex boxes. For p = 2, we construct explicit convex and nonconvex domains with intrinsic constant below the sharp scalar Sobolev constant. The nonconvex example gives an affirmative answer to the domain-existence question in Problem 5 of Mederski and Szulkin(Arch. Ration. Mech. Anal. 241 (2021), 1815-1842), while the convex example disproves the lower bound stated there for convex domains. Concerning their Problem 2, we obtain an implicit Newtonian representation of finite-energy solutions and exclude two special forms of solutions. These structural results do not yet provide an explicit ground state or an unconditional ground-state decay rate as asked in Problem 2.
ER -