In this paper, we describe the behavior of bounded energy finite solutions for certain nonlinear elliptic operators on a complete Riemannian manifold in terms of its p-harmonic boundary. We also prove that if two complete Riemannian manifolds are roughly isometric to each other, then their p-harmonic boundaries are homeomorphic to each other. In the case, there is a one to one correspondence between the sets of bounded energy finite solutions on such manifolds. In particular, in the case of the Laplacian, it becomes a linear isomorphism between the spaces of bounded harmonic functions with finite Dirichlet integral on the manifolds.
Bibliographical noteFunding Information:
★ This work was supported by grant No. R06-2002-012-01001-0(2002) from the Basic Research Program of the Korea Science & Engineering Foundation.
- p-harmonic boundary
- p-harmonic function
- Rough isometry