On the weak Harnack inequality for the parabolic p ( x )-Laplacian

In this paper a weak Harnack inequality for the parabolic p ( x )-Laplacian is established.

Remarks on parabolic De Giorgi classes

We make several remarks concerning properties of functions in parabolic De Giorgi classes of order p. There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex composition, and a logarithmic type estimate. Based on them, we are able to give new proofs of known properties. In particular, we prove local boundedness and local Hölder continuity of these functions via Moser’s ideas, thus avoiding De Giorgi’s heavy machinery. We also seize this opportunity to give a transparent proof of a weak Harnack inequality for nonnegative members of some super-class of De Giorgi, without any covering argument.

An improved Aleksandrov–Bakel’man–Pucci estimate for a second-order elliptic operator with unbounded drift

The classical Aleksandrov–Bakel’man–Pucci estimate (ABP estimate) for a second-order elliptic operator in nondivergence form is one of the fundamental tools for the bounds of subsolutions. Cabre improved the ABP estimate by replacing a constant factor, the diameter of a given domain, with a geometric character, which can be defined and finite for some unbounded domains. In the proof, Cabre used the Krylov–Safonov boundary weak Harnack inequality from Trudinger; thus, it is required that the first-order coefficients belong to a Lebesgue [Formula: see text]-integrable function space. Using a growth lemma from Safonov and an approximation method, we improve the result to Lebesgue [Formula: see text]-integrable first-order coefficients, which is optimal and coincides with the condition for the original ABP estimate.

Regularity and multiplicity results for fractional (p,q)-Laplacian equations

This paper deals with the study of the following nonlinear doubly nonlocal equation: [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary, [Formula: see text], with [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are parameters. Here [Formula: see text] and [Formula: see text] are sign-changing functions. We prove [Formula: see text] estimates, weak Harnack inequality and Interior Hölder regularity of the weak solutions of the above problem in the subcritical case [Formula: see text] Also, by analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to above convex–concave problem. In case of [Formula: see text], we show the existence of a solution.