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

Agnid Banerjee; Nicola Garofalo

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arXiv · 1202.5068

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

Agnid Banerjee·Nicola Garofalo

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Abstract

We construct viscosity solutions to the nonlinear evolution equation \eqref{p} 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 \cite{ES1} and \cite{SZ}. The pointwise properties of such solutions, namely the comparison principles, convergence of solutions as $p\to 1$, 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 ($p =2$).

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Agnid Banerjee, Nicola Garofalo. 2012-02-23. Gradient bounds and monotonicity of the energy for some nonlinear singular diffusion equations. https://arxiv.org/abs/1202.5068

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