Gradient bounds and monotonicity of the energy for some nonlinear singular diffusion equations
Abhishek BanerjeeNicola Garofalo
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 ($p=2$).
Indiana University Mathematics Journal
2013
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10.1512/iumj.2013.62.4969
10.1512/iumj.2013.62.4969
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Indiana Univ. Math. J. 62 (2013) 699 - 736
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