studies on the relationship of the structure of manifolds and p-harmonic functions

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P-Laplacian Δ_p(1<P<∞)is defined as operator acting on functions on Riemannian manifolds. The p-harmonic function u is defind by Δ_<p>u=div(|∇u|^<p-2> ∇u)=0. In the case of p=2, it becomes the usual harmonic map. We may consider the p-harmonic function the extension of harmonic function. In fact, the p-harmonic function is a critical point of the p-energy functional. Euler-Lagrange equation of it is that of the p-harmonic function. Because this equation is a nonlinear elliptic partial equation, it is hard to handle. We can define the p-Laplacian on graphs also, and define the p-harmonic function on graphs.We consider the spectrum of the p-Laplacian on graphs, p-harmonic morphisms between two graphs, and estimates for the solutions of p-Laplace equations on graphs. More precisely we prove a Ceeger type inequality and a Brooks type inequality for infinite graphs. We showed p-harmonic morphisms and horizontal conformal maps between two graphs are equivalent. We give some estimates for solutions of p-Laplace equations, which coincide with Green kernels in the case of p=2.Harmonic maps flow and p-harmonics flow are closely related to harmonic maps and p-harmonic maps.The stationary state of harmonic maps flow becomes the harmonic map, and the stationary state of p-harmonic maps flow becomes the p-harmonic map. But they do not necessarily converge, but blow-up of the solutions happen. We report this phenomena as research notes in Bulletin of Shikoku University.