$L^P$-Estimates And Polyharmonic Boundary Value Problems On The Sierpinski Gasket And Gaussian Free Fields On High Dimensional Sierpinski Carpet Graphs
| dc.contributor.author | Ugurcan, Baris | en_US |
| dc.contributor.chair | Strichartz, Robert Stephen | en_US |
| dc.contributor.committeeMember | Saloff-Coste, Laurent Pascal | en_US |
| dc.contributor.committeeMember | Muscalu, Florin Camil | en_US |
| dc.date.accessioned | 2015-01-07T20:56:59Z | |
| dc.date.available | 2019-08-19T06:01:32Z | |
| dc.date.issued | 2014-08-18 | en_US |
| dc.description.abstract | 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. | en_US |
| dc.identifier.other | bibid: 8793247 | |
| dc.identifier.uri | https://hdl.handle.net/1813/38759 | |
| dc.language.iso | en_US | en_US |
| dc.title | $L^P$-Estimates And Polyharmonic Boundary Value Problems On The Sierpinski Gasket And Gaussian Free Fields On High Dimensional Sierpinski Carpet Graphs | en_US |
| dc.type | dissertation or thesis | en_US |
| thesis.degree.discipline | Mathematics | |
| thesis.degree.grantor | Cornell University | en_US |
| thesis.degree.level | Doctor of Philosophy | |
| thesis.degree.name | Ph. D., Mathematics |
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