Sharp Hölder regularity for the inhomogeneous Trudinger’s equation

Abstract

Abstract We show that locally bounded solutions of the inhomogeneous Trudinger’s equation }2,$?> ∂ t | u | p − 2 u − d i v | ∇ u | p − 2 ∇ u = f ∈ L q , r , p > 2 , are locally Hölder continuous with exponent γ = min α 0 − , ( p q − n ) r − p q q ( p − 1 ) r , where α 0 denotes the optimal Hölder exponent for solutions of the homogeneous case. We provide a streamlined proof, using the full power of the homogeneity in the equation to develop the regularity analysis in the p-parabolic geometry, without any need of intrinsic scaling, as anticipated by Trudinger. The main difficulty in the proof is to overcome the lack of a translation invariance property.