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

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 ($p =2$)

A parabolic analogue of the higher-order comparison theorem of De Silva and Savin

We show that the quotient of two caloric functions which vanish on a portion of the lateral boundary of a $H^{k+ \alpha}$ domain is $H^{k+ \alpha}$ up to the boundary for $k \geq 2$. In the case $k=1$, we show that the quotient is in $H^{1+\alpha}$ if the domain is assumed to be space-time $C^{1, \alpha}$ regular. This can be thought of as a parabolic analogue of a recent important result in [DS1], and we closely follow the ideas in that paper. We also give counterexamples to the fact that analogous results are not true at points on the parabolic boundary which are not on the lateral boundary, i.e., points which are at the corner and base of the parabolic boundary

Riesz potentials and p-superharmonic functions in Lie groups of Heisenberg type

We prove a superposition principle for Riesz potentials of nonnegative continuous functions on Lie groups of Heisenberg type. More precisely, we show that the Riesz potential R_\alpha(\rho)(g) = \int_{\G} N(g^{-1} g')^{\alpha-Q} \rho(g') dg', \qquad 0<\alpha of a nonnegative function \rho\in C_0(\G) on a group \G of Heisenberg type is necessarily either $p$-subharmonic or $p$-superharmonic, depending on $p$ and $\alpha$. Here $N$ denotes the non-isotropic homogeneous norm on such groups, as introduced by Kaplan. This result extends to a wide class of nonabelian stratified Lie groups a recent remarkable superposition result of Lindqvist and Manfredi

Quantitative uniqueness for elliptic equations at the boundary of C1,DiniC^{1, Dini} domains

Based on a variant of the frequency function approach of Almgren, we establish an optimal upper bound on the vanishing order of solutions to variable coefficient Schr\"odinger equations at a portion of the boundary of a $C^{1,Dini}$ domain. Such bound provides a quantitative form of strong unique continuation at the boundary. It can be thought of as a boundary analogue of an interior result recently obtained by Bakri and Zhu for the standard Laplacian