Abstract
We extend to the parabolic setting some of the ideas originated with Xiao Zhong's proof in [31] of the Hölder regularity of $p-$harmonic functions in the Heisenberg group $\mathbb{H}^n$. Given a number $p\ge 2$, in this paper we establish the $C^{\infty}$ smoothness of weak solutions of a class of quasilinear PDE in $\mathbb{H}^n$ modeled on the equation $$?_t u = \sum_{i = 1}^{2n} X_i \bigg((1+|\nabla_0 u|^2)^{\frac{p-2}{2}} X_i u\bigg).$$
Highlights
In this paper we establish the C∞ smoothness of solutions of a certain class of quasilinear parabolic equations in the Heisenberg group Hn
The term regularized here refers to the fact that the non-linearity (1 + |∇0u|2) p−22 affects the ellipticity of the right hand side only when the gradient blows up, and not when it vanishes, presenting a weaker version of the singularity in the p−Laplacian
Given an open set Ω ⊂ Hn, we indicate with W 1,p(Ω) the Sobolev space associated with the penergy
Summary
In this paper we establish the C∞ smoothness of solutions of a certain class of quasilinear parabolic equations in the Heisenberg group Hn. The Caccioppoli inequalities needed to prove Theorem 3.1 will take up most of the section, and they all apply to a solution uε of the approximating equation (1.8) in a cylinder Q = Ω × (0, T ). 2n+1 ˆ t2 ˆ − 2 Aεi,ξj (x, ∇εuε)XlεXjεuεηβ+2|Zuε|β [Xiε, Xlε]uε i,j=1 t1 Ωt2 ˆ
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