Quasiconvexity and Lower Semicontinuity in the Riemannian Setting
Abstract
We introduce a notion of quasiconvexity for integrands defined on the bundle of linear maps between the tangent spaces of two smooth, connected and complete Riemannian manifolds (M,g) and (N,h), naturally extending the classical Euclidean notion. For continuous integrands, we prove that this Riemannian quasiconvexity is equivalent to the sequential lower semicontinuity of the associated functional F(u,Ω)=∫Ωf(du) dμg with respect to the weak* convergence in W1,∞(Ω,N), in the intrinsic sense adopted in the paper, for every bounded open subset Ω⊆ M. We then establish the Riemannian analogues of the Acerbi--Fusco theorems for Carathéodory integrands satisfying the corresponding growth conditions, for every 1leqslant pleqslant+∞, with respect to the weak and weak* notions of convergence considered here.
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BibTeX
@misc{corbisiero2026quasiconvexity,
title = {{Quasiconvexity and Lower Semicontinuity in the Riemannian Setting}},
author = {Aurora Corbisiero and Chiara Leone and Carlo Mantegazza},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02792},
eprint = {2611.02792},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02792},
}
RIS
TY - UNPB
TI - Quasiconvexity and Lower Semicontinuity in the Riemannian Setting
AU - Aurora Corbisiero
AU - Chiara Leone
AU - Carlo Mantegazza
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02792
UR - https://arxiv.org/abs/2611.02792
AB - We introduce a notion of quasiconvexity for integrands defined on the bundle of linear maps between the tangent spaces of two smooth, connected and complete Riemannian manifolds (M,g) and (N,h), naturally extending the classical Euclidean notion. For continuous integrands, we prove that this Riemannian quasiconvexity is equivalent to the sequential lower semicontinuity of the associated functional F(u,Ω)=∫_Ωf(du) dμ_g with respect to the weak^* convergence in W^1,∞(Ω,N), in the intrinsic sense adopted in the paper, for every bounded open subset Ω⊆ M. We then establish the Riemannian analogues of the Acerbi--Fusco theorems for Carathéodory integrands satisfying the corresponding growth conditions, for every 1leqslant pleqslant+∞, with respect to the weak and weak^* notions of convergence considered here.
ER -