University of Isfahan Mathematics and Society 2345-6493 4 1 2019 05 22 Level set method for motion by mean curvature Level set method for motion by mean curvature 7 17 23953 10.22108/msci.2019.113074.1296 FA Mehran Aminian Vali-e-Asr University of Rafsanjan, Rafsanjan Mehran Namjoo Vali-e-Asr University of Rafsanjan, Rafsanjan 0000-0001-5949-6766 Journal Article 2018 09 26 This paper is a translation of the the following paper into Persian:<br />[T. H. Colding and W. P. Minicozzi II, Level Set Method For Motion by Mean Curvature, <em>Notices of the AMS</em>, <strong>63 </strong>no. 10 (2016) 1148–1153.]<br /> <br /> <span class="fontstyle0">Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to one of the classical degenerate nonlinear second-order differential equations on Euclidean space. One naturally wonders, “What is the regularity of solutions?” A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid situations that have a simple geometric interpretation. The proof weaves together analysis and geometry. Without deeply understanding the underlying geometry, it is impossible to prove fine analytical properties.</span> This paper is a translation of the the following paper into Persian:<br />[T. H. Colding and W. P. Minicozzi II, Level Set Method For Motion by Mean Curvature, <em>Notices of the AMS</em>, <strong>63 </strong>no. 10 (2016) 1148–1153.]<br /> <br /> <span class="fontstyle0">Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to one of the classical degenerate nonlinear second-order differential equations on Euclidean space. One naturally wonders, “What is the regularity of solutions?” A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid situations that have a simple geometric interpretation. The proof weaves together analysis and geometry. Without deeply understanding the underlying geometry, it is impossible to prove fine analytical properties.</span> https://math-sci.ui.ac.ir/article_23953_e3e6bdaa31e2225b3f23cd23f4a810d4.pdf