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The fattening phenomenon for level set solutions of the mean curvature flow

University of British Columbia

Level set solutions are an important class of weak solutions to the mean curvature flow which allow the flow to be extended past singularities. Unfortunately, when singularities do develop it is possible for the Hausdorff dimension of the level set solution to increase. This behaviour is referred to as the fattening phenomenon. The purpose of this thesis is to discuss this phenomenon and to provide concrete examples, focusing especially on its relation to the uniqueness of smooth solutions. We first discuss the definition of level set solutions in arbitrary codimension, due to Ambrosio and Soner. We then prove some technical results about distance solutions, a type of set-theoretic subsolution to level set solutions. These include a new method of gluing together distance solutions. Next, we present several known results on the fattening phenomenon in the context of distance solutions. Finally, we provide a new example by proving that fattening occurs when immersed curves in ℝ³ develop self-intersections.

Text

2017-04-19

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61267

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/