An Intrinsic Sobolev Trace Theory for Riemannian Vector Bundles with Applications to Quasilinear Elliptic Equations
Abstract
Let E→ M be a finite-rank Riemannian vector bundle over a compact Riemannian manifold with nonempty smooth boundary, equipped with a compatible connection. We develop an intrinsic trace theory for Sobolev sections defined through weak covariant derivatives. For every integer m≥1 and every 1≤ p<∞, we construct continuous trace operators for the covariant derivatives up to order m−1 and characterize the Sobolev space with homogeneous Dirichlet boundary conditions as the kernel of the corresponding trace operator. As an application, we establish the existence of ground state solutions for a class of quasilinear elliptic equations on Riemannian vector bundles with generalized (p,q)-growth. The proposed framework unifies several important geometric operators, including the Bochner p-Laplacian, the (p,q)-Bochner Laplacian, and prescribed mean curvature-type operators.
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BibTeX
@misc{velazquezmendoza2026intrinsic,
title = {{An Intrinsic Sobolev Trace Theory for Riemannian Vector Bundles with Applications to Quasilinear Elliptic Equations}},
author = {Carlos Daniel Velázquez-Mendoza and María de los Ángeles Sandoval-Romero and Romulo Diaz Carlos},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26744},
eprint = {2609.26744},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26744},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - An Intrinsic Sobolev Trace Theory for Riemannian Vector Bundles with Applications to Quasilinear Elliptic Equations
AU - Carlos Daniel Velázquez-Mendoza
AU - María de los Ángeles Sandoval-Romero
AU - Romulo Diaz Carlos
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26744
UR - https://arxiv.org/abs/2609.26744
AB - Let E→ M be a finite-rank Riemannian vector bundle over a compact Riemannian manifold with nonempty smooth boundary, equipped with a compatible connection. We develop an intrinsic trace theory for Sobolev sections defined through weak covariant derivatives. For every integer m≥1 and every 1≤ p<∞, we construct continuous trace operators for the covariant derivatives up to order m−1 and characterize the Sobolev space with homogeneous Dirichlet boundary conditions as the kernel of the corresponding trace operator. As an application, we establish the existence of ground state solutions for a class of quasilinear elliptic equations on Riemannian vector bundles with generalized (p,q)-growth. The proposed framework unifies several important geometric operators, including the Bochner p-Laplacian, the (p,q)-Bochner Laplacian, and prescribed mean curvature-type operators.
ER -