Regularity properties of viscosity solutions for fully nonlinear equations on the model of the anisotropic p →-Laplacian

Abstract

We consider the pseudo-p-Laplacian operator:We prove interior regularity results for the viscosity (resp. weak) solutions in the unit ball B 1 of∆ p u = (p − 1)f for f ∈ C(B 1 ) (resp. f ∈ L ∞ (B 1 )) : Firstly the Hölder local regularity for any exponent γ < 1, recovering in that way a known result about weak solutions. In a second time we prove the Lipschitz local regularity.

Lipschitz regularity for orthotropic functionals with nonstandard growth conditions

“…Incidentally, we point out that this is the same assumption as in the aforementioned paper [13], which uses however different techniques.…”

Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators

“…In [19], [11] we considered viscosity solutions for the fully non linear extension of the pseudop-Laplacian, say the case where in (1.4), the left hand side is replaced by −F (Θ α (∇u)D 2 uΘ α (∇u)), and α > 0. More general anisotropic fully non linear degeneracy is treated in [20]. In the variational case, one important and recent result can be found in [7].…”

Regularity under general and <inline-formula><tex-math id="M1">\begin{document}$ p,q- $\end{document}</tex-math></inline-formula> growth conditions

“…. , n. Since weak solutions which are continuous are also viscosity solutions, then Françoise Demengel [37] (see also [9]) proved that, under the condition q − p < 1, then every continuous weak solution is locally Lipschitz continuous too. Note that, also in this context, a-priori locally bounded solutions are considered and the regularity results of Section 5 apply.…”