Sharp Hölder regularity for the inhomogeneous Trudinger’s equation

We show that locally bounded solutions of the inhomogeneous Trudinger’s equation ∂t (|u|p−2u) − div|∇u|p−2∇u = f ∈ Lq,r, p > 2, are locally Hölder continuous with exponent γ = min { α−0(pqq−(pn−)r1)−rpq} , where α0 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.