Some Properties of Laplacians on Fractals

Kigami has defined an analog of the Laplacian on a class of self-similar fractals, including the familiar Sierpinski gasket. We study properties of this operator. We show that there is a maximal principle for solutions of certain nonlinear equations of the formΔu(x)=F(x, u(x)). We discuss the extension of the Laplacian to non-compact fractal blow-ups, and show that it is essentially self-adjoint, and we prove an analog of Liouvilleʹs theorem in some cases. We also give an explicit algorithm for solving the Dirichlet problem for certain domains in the Sierpinski gasket and give a characterization of all harmonic functions on those domains.