On Sobolev spaces and density theorems on Finsler manifolds

Document Type : Original Article


Department of Mathematics and Computer Science, Amirkabir University of Technology, 424, Hafez Ave., Tehran 15914, Iran


Here, a natural extension of Sobolev spaces is defined for a Finsler structure $F$ and it is shown that the set of all real $C^{\infty}$ 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 $\Delta u=f$ can be approximated by $C^{\infty}$ functions with compact support on $M$. Moreover, let $W\subseteq M$ be a regular domain with the $C^r$ boundary $\partial W$, then the set of all real functions in $C^r(W)\cap C^0(\overline{W})$ is dense in $H^p_k(W)$, where $k\leq r$. Finally, several examples are illustrated and sharpness of the inequality $k\leq r$ is shown.


Main Subjects

[1] R. Adams, Sobolev spaces, Academic press, New York, 1975.
[2] H. Akbar-Zadeh, Initiation to Global Finslerian Geometry, North- Holland Mathematical Library, 2006.
[3] T. Aubin, Some nonlinear problems in Riemannian geometry, Springer-Verlag, 1988.
[4] S. Azami, A. Razavi, Existence and uniqueness for a solution of Ricci flow on Finsler manifolds, Int. J. of Geom. Meth. in Mod. Phy., 10(3) (2013) 1-21.
[5] D. Bao, S. S. Chern, Z. Shen, Riemann-Finsler geometry, Springer-Verlag, 2000.
[6] D. Bao, B. Lackey, A Hodge decomposition theorem for Finsler spaces, C. R. Acad. Sci. Paris S´er. I Math., 323(1) (1996) 51-56.
[7] B. Bidabad, On compact Finsler spaces of positive constant curvature C. R. Acad. Sci. Paris S´er. I Math., 349 (2011) 1191-1194.
[8] B. Bidabad, A. Shahi, Harmonic vector fields on Finsler manifolds, C. R. Acad. Sci. Paris S´er. I Math., 354 (2016) 101-106.
[9] B. Bidabad, A. Shahi, M. Yar Ahmadi, Deformation of Cartan curvature on Finsler manifolds, Bull. Korean Math. Soc. 54(6) (2017) 2119-2139.
[10] B. Bidabad, M. Yar Ahmadi, Convergence of Finslerian metrics under Ricci flow, Sci. China Math. 59(4) (2016) 741-750.
[11] Y. Ge, Z. Shen, Eigenvalues, and eigenfunctions of metric measure manifolds, Proc. London Math. Soc., 82(3) (2001) 725-746.
[12] Q. He, Y. Shen, On Bernstein type theorems in Finsler spaces with the volume form induced from the projective sphere bundle, Proc. Amer. Math. Soc., 134(3) (2006) 871-880.
[13] M. Jim´enez-Sevilla, L. Sanchez-Gonzalez, On some problems on smooth approximation and smooth extension of Lipschitz functions on Banach-Finsler manifolds, Nonlinear Anal. 74(11) (2011) 3487-3500.
[14] A. Krist´aly, I. Rudas, Elliptic problems on the ball endowed with Funk-type metrics, Nonlinear Anal., 119 (2015) 199-208.
[15] S. Lakzian, Differential Harnack estimates for positive solutions to heat equation under Finsler-Ricci flow, Pacific J. Math., 278(2) (2015) 447-462
[16] H. Mosel, S. Winkelmann, On weakly harmonic maps from Finsler to Riemannian manifolds, Ann. I. H. Poincar´e, 26 (2009) 39-57.
[17] S. B. Myers, Algebras of differentiable functions, Proc. Amer. Math. Soc., 5 (1954) 917-922.
[18] S. Ohta, Nonlinear geometric analysis on Finsler manifolds, European Journal of Math., 3(4) (2017) 916-952.
[19] Z. Shen, Lectures on Finsler geometry, World Scientific, 2001.
[20] N. Winter, On the distance function to the cut locus of a submanifold in Finsler geometry, Ph.D. thesis, RWTH Aachen University, (2010).
[21] Y. Yang, Solvability of some elliptic equations on Finsler manifolds, math.pku.edu.cn preprint, 1-12.