Abstract
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.
Highlights
De Giorgi classes consist of Sobolev functions in an open set ⊂ RN satisfying a family of energy estimates, i.e., u ∈ Wl1o,cp( ) and for some γ > 0, ˆ K | D (u (y) − k)±|p dx ≤ (R γ − )p |(u − k)±|p dxK R (y) for all k ∈ R, and any pair of concentric cubes K (y) ⊂ K R(y) in
A probably even more striking discovery was made by DiBenedetto and Trudinger in [6] that nonnegative members of De Giorgi classes
In addition to De Giorgi’s techniques, the main new input of [6] includes realization of pointwise lower bound of nonnegative members in De Giorgi classes with a power-like dependence on the measure distribution of their positivity
Summary
De Giorgi classes consist of Sobolev functions in an open set ⊂ RN satisfying a family of energy estimates, i.e., u ∈ Wl1o,cp( ) and for some γ > 0, ˆ. The feature of Moser’s approach is twofold; on the one hand, it simplifies the original proof of De Giorgi and gives a more intuitive method; on the other hand, it keeps referring to the equation This latter point renders a question on whether we could use Moser’s idea in [16] to show the Hölder regularity for functions in De Giorgi classes, where no equations are at our disposal. 5, we give a proof of Hölder regularity for members of certain parabolic De Giorgi classes, via Moser’s ideas, avoiding De Giorgi’s heavy machinery. This parallels the result for the elliptic De Giorgi classes in [10].
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