C1 regularity of orthotropic p-harmonic functions in the plane

On the Lipschitz character of orthotropic p-harmonic functions

“…(1) one word about the assumption p ≥ 2: as explained in [1] and [2], when δ 1 = · · · = δ N = 0 the subquadratic case 1 < p < 2 is simpler in a sense. In this case, the desired Lipschitz regularity can be inferred from [ (2) in [1,Main Theorem], local minimizers were proven to be C 1 , in the two-dimensional case, for 1 < p < ∞ and when δ 1 = · · · = δ N = 0. We also refer to the very recent paper [14], where a modulus of continuity for the gradient of local mimizers is exhibited.…”

“…However, as explained in [1] and [2], equation (1.2) is much more degenerate. Consequently, as for the regularity of ∇u (i.e.…”

Regularity of the derivatives of p-orthotropic functions in the plane for 1 < p < 2

“…Proof. The proof of the Lipschitz bound can be found in [4] while (2.6) appears in [1]. We provide details for completeness.…”

“…The equation is singular when either one of the derivatives vanishes, and does not fall into the category of equations with p-Laplacian structure. It was proved by Bousquet and Brasco in [1] that weak solutions of (1.1) for 1 < p < ∞ are C 1 (Ω). A simple proof which gives a logarithmic modulus of continuity for the derivatives is contained in [6] for the case p ≥ 2.…”

Regularity for an anisotropic equation in the plane

“…Our object is the continuity of the gradient ∇u = (u x 1 , u x 2 ) in the plane. The recent work [1] of P. Bousquet in Ω, with only one exponent 1 < p < ∞. They proved that u ∈ C 1 loc (Ω).…”