\(p\)-energy and \(p\)-harmonic functions on Sierpinski gasket type fractals
The classical \(p\)-energy of a function \(u\) is the integral of \(| \nabla u| ^p\), for the gradient \(\nabla\) and \(1< p<\infty\). The techniques of \textit{J. Kigami} [Analysis on fractals. (Cambridge Tracts in Math. 143, Cambridge University Press, Cambridge) (2001; Zbl 0998.28004)] for the case \(p=2\) are extended. The fractal is approximated by a sequence of graphs endowed with their discrete \(p\)-energies. A nonlinear renormalization problem is solved to find an appropriate scaling factor and a corresponding self-similar discrete \(p\)-energy. The construction of the limiting model mimics the case \(p=2\). The existence result relies on Schauder's fixed point theorem but the uniqueness remains open. A function is termed \(p\)-harmonic when it minimizes the \(p\)-energy subject to boundary conditions. Its uniqueness is again only conjectured. Numerical approximations for the Sierpinski gasket are presented. The existence proof is generalized to Peirone's ``weakly completely symmetric fractals'' [Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 3, No. 2, 431--460 (2000; Zbl 0958.31005)].
scientific article; zbMATH DE number 2057479