On Sobolev spaces and density theorems on Finsler manifolds

2020/02/19 by Behroz Bidabad, Alireza Shahi
Mathematics · #math.DG #msc:53C60 #msc:46E35

paper · pdf · doi:10.22060/ajmc.2018.3039

published as On Sobolev spaces and density theorems on Finsler manifolds, AUT J. Math. Com. 1 2020, 37-45 · Published in AUT Journal of Math. Comp. arXiv admin note: text overlap with arXiv:1310.8027

arxiv created 2020/02/19 · arxiv updated 2020/02/21

Abstract

Here, a natural extension of Sobolev spaces is defined for a Finsler structure F and it is shown that the set of all real C^∞ functions with compact support on a forward geodesically complete Finsler manifold (M, F), is dense in the extended Sobolev space H₁^p (M). As a consequence, the weak solutions u of the Dirichlet equation Δu=f can be approximated by C^∞ functions with compact support on M. Moreover, let W ⊂ M be a regular domain with the C^r boundary ∂ W, then the set of all real functions in C^r (W) ∩ C⁰ ( W) is dense in H_k^p (W), where k≤ r. Finally, several examples are illustrated and sharpness of the inequality k≤ r is shown.

Citations

• On Sobolev spaces and density theorems on Finsler manifolds