Regularity for an anisotropic equation in the plane

Abstract

We present a simple proof of the C 1 regularity of p-anisotropic functions in the plane for 2 ≤ p < ∞. We achieve a logarithmic modulus of continuity for the derivatives. The monotonicity (in the sense of Lebesgue) of the derivatives is used. The case with two exponents is also included.

On the Lipschitz character of orthotropic p-harmonic functions

“…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. We do not know whether such a result still holds in higher dimensions;…”

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

“…The latter relies on a lemma on the oscillation of monotone functions due to Lebesgue [5] and the fact that derivatives of solutions are monotone (in the sense of Lebesgue). The purpose of this work is to extend this result to the case 1 < p < 2 employing methods developed in [6]. We obtain the following:…”

“…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. The latter relies on a lemma on the oscillation of monotone functions due to Lebesgue [5] and the fact that derivatives of solutions are monotone (in the sense of Lebesgue).…”

The asymptotic behavior for anisotropic parabolic p‐Laplacian equations

“…Recently, the anisotropic elliptic or parabolic equations attract much attention, in other words, people turn to study of the anisotropic p ‐Laplacian equations, where the anisotropic operator is where is a constant vector, , we refer to [2, 6, 9–13, 15, 16, 23, 25, 26, 29, 35] and their references. In especially, we would like to mention the result of Agnese DI Castro, Eugenio Montefusco [12] who considered the following anisotropic quasilinear degenerate elliptic equations where Ω is a bounded subset of , and assume are ordered.…”