"Hot Spots" Conjecture And Fractal Interpolation Functions On P.c.f. Self-similar Sets

We studied two problems in this thesis.1. "Hot spots" conjecture. The "hot spots" conjecture was posed by J.Rauch in 1974, which focus on domains in Euclidean space. Using spectral decimation, Ruan proved that the "hot spots"conjecture holds on Sierpinski gaskets in 2012. In this paper, we will continue the work of "hot spots" conjecture on p.c.f.self-similar sets. We will prove the conjecture holds on higher dimen-sional Sierpinski gasket, then we show that it does not hold on hexagasket. It follows that the "hot spots" conjecture does not hold on p.c.f.self-similar sets.2. Fractal interpolation functions (FIFs). We first characterize the finiteness of FIFs on p.c.f.self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling factors on Sierpinski gasket (SG). As an application, we prove that the solution of the following Dirichlet problem on SG is an FIF with uniform vertical scaling factor 5/1:where qi, i = 1,2,3, are boundary points of SG, a1,a2,a3,?? R. In the last part, by using properties of harmonic function, we discusses the min-max property of fractal interpolation function on SG with uniform vertical scaling factor, and presents a necessary and sufficient condition such that the function has the same range with its basic function.