Second-Order Regularity for Degenerate Parabolic Quasi-Linear Equations in the Heisenberg Group

Abstract

In the Heisenberg group Hn, we obtain the local second-order HWloc2,2-regularity for the weak solution u to a class of degenerate parabolic quasi-linear equations ∂tu=∑i=12nXiAi(Xu) modeled on the parabolic p-Laplacian equation. Specifically, when 2≤p≤4, we demonstrate the integrability of (∂tu)2, namely, ∂tu∈Lloc2; when 2≤p3, we demonstrate the HWloc2,2-regularity of u, namely, XXu∈Lloc2. For the HWloc2,2-regularity, when p≥2, the range of p is optimal compared to the Euclidean case.

Keywords

• parabolic quasi-linear equation
• second-order regularity
• Heisenberg group
• parabolic <i>p</i>-Laplacian equation
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