Supercaloric functions for the porous medium equation in the fast diffusion case

We study a generalized class of supersolutions, so-called supercaloric functions to the porous medium equation in the fast diffusion case. Supercaloric functions are defined as lower semicontinuous functions obeying a parabolic comparison principle. We prove that bounded supercaloric functions are weak supersolutions. In the supercritical range, we show that unbounded supercaloric functions can be divided into two mutually exclusive classes dictated by the Barenblatt solution and the infinite point-source solution, and give several characterizations for these classes. Furthermore, we study the pointwise behavior of supercaloric functions and obtain connections between supercaloric functions and weak supersolutions.Comment: Corrected typographical errors and made minor notational adjustment

On the definition of solution to the total variation flow

We show that the notions of weak solution to the total variation flow based on the Anzellotti pairing and the variational inequality coincide under some restrictions on the boundary data. The key ingredient in the argument is a duality result for the total variation functional, which is based on an approximation of the total variation by area-type functionals

POTENTIAL ESTIMATES IN PARABOLIC OBSTACLE PROBLEMS

Abstract. For parabolic obstacle problems with quadratic growth, we give pointwise estimates both for the solutions and their gradients in terms of potentials of the given data. As applications, we derive Lorentz space estimates if the data satisfies the corresponding Lorentz space regularity. Moreover, we discuss a borderline case in the regularity theory, the question of boundedness and continuity of the gradients as well as of the solutions itself