882 research outputs found
A certain critical density property for invariant Harnack inequalities in H-type groups
We consider second order linear degenerate-elliptic operators which are
elliptic with respect to horizontal directions generating a stratified algebra
of H-type. Extending a result by Guti\'errez and Tournier for the Heisenberg
group, we prove a critical density estimate by assuming a condition of
Cordes-Landis type. We then deduce an invariant Harnack inequality for the
non-negative solutions from a result by Di Fazio, Guti\'errez, and Lanconelli.Comment: 13 page
Wiener criterion for X-elliptic operators
In this note we prove a Wiener criterion of regularity of boundary points for
the Dirichlet problem related to -elliptic operators in divergence form
enjoying the doubling condition and the Poincar\'e inequality. As a step
towards this result, we exhibit some other characterizations of regularity in
terms of the capacitary potentials. Finally, we also show that a cone-type
criterion holds true in our setting
Harnack's inequality for a class of non-divergent equations in the Heisenberg group
We prove an invariant Harnack's inequality for operators in non-divergence
form structured on Heisenberg vector fields when the coefficient matrix is
uniformly positive definite, continuous, and symplectic. The method consists in
constructing appropriate barriers to obtain pointwise-to-measure estimates for
supersolutions in small balls, and then invoking the axiomatic approach from
[DGL08] to obtain Harnack's inequality
Wiener-Landis criterion for Kolmogorov-type operators
We establish a necessary and sufficient condition for a boundary point to be
regular for the Dirichlet problem related to a class of Kolmogorov-type
equations. Our criterion is inspired by two classical criteria for the heat
equation: the Evans-Gariepy's Wiener test, and a criterion by Landis expressed
in terms of a series of caloric potentials
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