PreprintarXiv licencearXiv
Sharp Weighted Sobolev Trace Inequalities on Convex Cones and Rigidity of Extremals
Abstract
We study critical weighted Sobolev trace inequalities on convex cones endowed with positively homogeneous weights. Using mass transportation, we first establish the continuity of the critical trace embedding under general homogeneity and concavity assumptions on the weight. We then identify a compatibility condition between the weight, the cone, and a distinguished direction that makes the transport argument sharp. In this regime, we determine the optimal constant and prove the existence of extremals. Finally, we obtain a complete classification of all nontrivial extremals modulo translation in the lineality space.
Related
- Extremals for Sharp Trace Gagliardo--Nirenberg--Sobolev Inequalities on Convex Cones
- Extremals for singular Onofri type inequality in weighted Sobolev spaces
- Critical Sobolev Inequalities and Extremals for a Partially Fractional Grushin Operator
- Large-time asymptotics for general cross-diffusion systems and novel convex Sobolev inequalities
- On improved Hardy-type inequalities in Log-Weighted Sobolev Spaces
BibTeX
@misc{do2026sharp,
title = {{Sharp Weighted Sobolev Trace Inequalities on Convex Cones and Rigidity of Extremals}},
author = {João Marcos do Ó and Lucas de Melo Pontes e Silva},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24931},
eprint = {2610.24931},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24931},
}
RIS
TY - UNPB
TI - Sharp Weighted Sobolev Trace Inequalities on Convex Cones and Rigidity of Extremals
AU - João Marcos do Ó
AU - Lucas de Melo Pontes e Silva
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.24931
UR - https://arxiv.org/abs/2610.24931
AB - We study critical weighted Sobolev trace inequalities on convex cones endowed with positively homogeneous weights. Using mass transportation, we first establish the continuity of the critical trace embedding under general homogeneity and concavity assumptions on the weight. We then identify a compatibility condition between the weight, the cone, and a distinguished direction that makes the transport argument sharp. In this regime, we determine the optimal constant and prove the existence of extremals. Finally, we obtain a complete classification of all nontrivial extremals modulo translation in the lineality space.
ER -