Harnack's Estimates: Positivity and Local Behavior of Degenerate and Singular Parabolic Equations

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Local properties of non–negative solutions to some doubly non–linear degenerate parabolic equations

In the present paper we study the local behavior of non-negative weak solutions of a wide class of doubly non linear degenerate parabolic equations. We show, in particular, some lower pointwise estimates of such solutions in terms of suitable sub-potentials (dictated by the structure of the equation) and an alternative form of the Harnack inequality

Self-improving property of degenerate parabolic equations of porous medium-type

We show that the gradient of solutions to degenerate parabolic equations of porous medium-type satisfies a reverse H"older inequality in suitable intrinsic cylinders. We modify the by-now classical Gehring lemma by introducing an intrinsic Calder'on-Zygmund covering argument, and we are able to prove local higher integrability of the gradient of a proper power of the solution $u$

A Wiener-Type Condition for Boundary Continuity of Quasi-Minima of Variational Integrals

A Wiener-type condition for the continuity at the boundary points of Q-minima, is established, in terms of the divergence of a suitable Wiener integral

Continuity of the Saturation in the Flow of Two Immiscible Fluids in a Porous Medium

We consider a weakly coupled system, consisting of an elliptic equation and a degenerate parabolic equation; such a system arises in the theory of flow of immiscible fluids in a porous medium. The unknown functions u and v and the equations they satisfy, represent the pressure and the saturation respectively, subject to Darcy’s law and the Buckley-Leverett coupling. Due to the empirical nature of these laws no determination is possible on the structure of the degeneracy exhibited by the system. It is established that the saturation is a locally continuous function in its space-time domain of definition, irrespective of the nature of the degeneracy of the principal part of the system