On Sobolev spaces and density theorems on Finsler manifolds

Document Type : Original Article

Authors

1 Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic)

2 Faculty of Mathematics and computer science, Amirkabir University of Technology

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 H p 1 (M). As a consequence, the weak solutions u of the Dirichlet equation ∆u = 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) ∩ C 0 (W) is dense in H p k (W), where k ≤ r. Finally, several examples are illustrated and sharpness of the inequality k ≤ r is shown

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