First PositionPostdoctoral Fellowship, University of Western Ontario
Dissertation$L^P$-Estimates And Polyharmonic Boundary Value Problems On The Sierpinski Gasket And Gaussian Free Fields On High Dimensional Sierpinski Carpet Graphs
We define a suitable trace space on the set X halving the Sierpinski Gasket, then we prove Lp -estimates for p > 1 for the restriction operator on domLp [INCREMENT](SG). We also construct a right inverse to the restriction operator, that is the extension operator, and provide similar Lp -estimates. Then, we consider the polyharmonic boundary value problem which involves finding a biharmonic function with prescribed values and Laplacian values on the bottom line (identified with the interval) and top vertex of the SG. After constructing a suitable orthogonal basis of piecewise biharmonic splines, we express the solution to the BV P in terms of the Haar expansion coefficients of the prescribed data and this basis. After constructing a Sobolev type space on SG, which is analogous to the H 2 -Sobolev space in classical analysis, we prove how smoothness of the prescribed data is reflected in the smoothness of the solution to the BV P . In the second part of the thesis, we focus on Gaussian Free Fields on High dimensions Sierpinski Carpet graphs. We assume that a "hard wall" is imposed at height zero so that the field stays positive everywhere. Our first result, in the second part of the thesis, is a large deviation type estimate which identifies the rate of exponential decay for P(Ω+N ), namely the probability that the field stays positive. Then, in our second V theorem we prove the leading-order asymptotics for the local sample mean of the free field above the hard wall on any transient Sierpinski carpet graph.