Note on an elementary inequality and its application to the regularity of p-harmonic functions

Abstract

We study the Sobolev regularity of \(p\)-harmonic functions. We show that \(|Du|^{\frac{p-2+s}{2}}Du\) belongs to the Sobolev space \(W^{1,2}_{\operatorname{loc}}\), \(s>-1-\frac{p-1}{n-1}\), for any \(p\)-harmonic function \(u\). The proof is based on an elementary inequality.

On the second-order regularity of solutions to the parabolic p-Laplace equation

“…For the homogeneous problem, our main result, Theorem 1.1 below, is a generalization of [21, Theorem 1.1] and [41,Theorem 1.1]. These known results concern the p-Laplace equation ∆ p u = 0, and they yield W 1,2 locregularity of |∇u| β ∇u for β > −1…”

“…Now the desired estimate follows from Lemma 4.5. Range (5.2) in Proposition 5.1, is optimal in the following sense: In the elliptic case, [10] and [25], the best known range is s > −1 − p−1 n−1 . On the other hand, Example 5.1 below shows that in the parabolic case we cannot hope to reach any better range than s > γ + 1 − p. A counterexample of this type was used in [10, Section 1.3] for the standard p-parabolic equation.…”

“…Manfredi and Weitsman [36] proved that D 2 u ∈ L 2 loc (Ω) on the range 1 < p < 3 + 2 n−2 . These results were later developed by Dong, Peng, Zhang and Zhou [21] and the second author [41] to cover the W 1,2 loc -regularity of a family of vector fields |∇u| β ∇u, where β > −1 + (n−2)(p−1) 2(n−1) . For the non-homogeneous problem ∆ p u = f , a natural question to ask is the following: Given the regularity of f , can we trade the divergence operator div with the full derivative D?…”

“…which holds for any smooth function u as shown by Sarsa in [25]. Curiously, in [11] it was sufficient to use the above inequality in a simpler form just estimating (|Du| 2 ∆u − ∆ ∞ u) 2 ≥ 0 on the right hand side.…”

Jacobian determinants for (nonlinear) gradient of planar $\infty$-harmonic functions and applications

“…In the literature, the interior regularity of p-harmonic functions in any dimension has been extensively studied. See [35,36,15,13,26,8,34,29,28,14,32].…”