article

Title: Gradient bounds and monotonicity of the energy for some nonlinear singular diffusion equations

Authors: Agnid Banerjee and Nicola Garofalo

Issue: Volume 62 (2013), Issue 2, 699-736

Abstract:

$We construct viscosity solutions to the nonlinear evolution equation (1.4) below, which generalizes the motion of level sets by mean curvature (the latter corresponds to the case $p=1$) using the regularization scheme as in [L.\:C. Evans and J. Spruck, \textit{Motion of level sets by mean curvature. I}, J. Differential Geom. \textbf{33} (1991), no. 3, 635--681] and [P. Sternberg and W.\:P. Ziemer, \textit{Generalized motion by curvature with a Dirichlet condition}, J. Differential Equations \textbf{114} (1994), no. 2, 580--600]. The pointwise properties of such solutions---namely, the comparison principles, convergence of solutions as $p\to1$, large-time behavior, and unweighted energy monotonicity---are studied. We also prove a notable monotonicity formula for the weighted energy, thus generalizing Struwe's famous monotonicity formula for the heat equation ($p=2$).$