455 research outputs found
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 ) 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 , 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 ()
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 domain is up to
the boundary for . In the case , we show that the quotient is in
if the domain is assumed to be space-time
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 -subharmonic or -superharmonic, depending on and . Here 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 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
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
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