Growth theorems and Harnack inequality for second order parabolic equations (Q2767867)

The authors prove the interior Harnack inequality for solutions of both divergence (D) and non-divergence (ND) equations by using growth theorems as a common background. These growth theorems control the behavior of (sub- super-) solutions of second-order elliptic or parabolic equations in terms of the Lebesgue measure of areas in which solutions are positive or negative.NEWLINENEWLINENEWLINEThese theorems, which are improvements of growth theorems from the book by \textit{E. M. Landis} [Second-order equations of elliptic and parabolic type, Translations of Mathematical Monographs, 171, Providence, RI: AMS (1998; Zbl 0895.35001)], allow the authors to derive the interior Harnack inequality as consequence. They prove three such theorems assuming that all functions (coefficients and solutions) are smooth enough to treat the cases (D) and (ND) simultaneously. Then they get rid of extra smoothness assumptions by means of standard approximation procedures which are briefly discussed at the end of the paper (see Remark 6.1). These procedures, and also the minimal smoothness assumptions of coefficients and solutions, are different in the cases (D) and (ND). In both cases it is important to have appropriate estimates for the solutions with constants depending only on the structural parameters and not depending on the ``additional smoothness''.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00020].