Mean value and Harnack inequalities for degenerate parabolic equations
On the second order derivatives of convex functions on the Heisenberg group
In the euclidean setting the celebrated Aleksandrov-Busemann-Feller theorem states that convex functions are a.e. twice differentiable. In this paper we prove that a similar result holds in the Heisenberg group, by showing that every continuous –convex function belongs to the class of functions whose second order horizontal distributional derivatives are Radon measures. Together with a recent result by Ambrosio and Magnani, this proves the existence a.e. of second order horizontal derivatives for...
Classical, viscosity and average solutions for PDE’s with nonnegative characteristic form
We compare several definitions of weak solutions to second order partial differential equations with nonnegative characteristic form.
Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators
In this Note we prove a two-weight Sobolev-Poincaré inequality for the function spaces associated with a Grushin type operator. Conditions on the weights are formulated in terms of a strong weight with respect to the metric associated with the operator. Roughly speaking, the strong condition provides relationships between line and solid integrals of the weight. Then, this result is applied in order to prove Harnack's inequality for positive weak solutions of some degenerate elliptic equations....
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