Global regularity for a class of fully nonlinear PDEs with unbalanced variable degeneracy

Abstract

We establish the existence and sharp global regularity results (C0,γ$C^{0, \gamma }$, C0,1$C^{0, 1}$ and C1,α$C^{1, \alpha }$ estimates) for a class of fully nonlinear elliptic Partial Differential Equations (PDEs) with unbalanced variable degeneracy. In a precise way, the degeneracy law of the model switches between two different kinds of degenerate elliptic operators of variable order, according to the null set of a modulating function frakturafalse(·false)⩾0$\mathfrak {a}(\cdot )\geqslant 0$. The model case in question is given by -0.16em-0.16em{false|Dufalse|pfalse(xfalse)+fraktura(x)false|Dufalse|qfalse(xfalse)scriptMλ,Λ+false(D2ufalse)=ffalse(xfalse)inΩu(x)=g(x)on∂Ω,$$\begin{equation*} \!\!{\left\lbrace \! \def\eqcellsep{&}\begin{array}{rcl} {\left[|Du|^{p(x)}+\mathfrak {a}(x)|Du|^{q(x)}\right]}\mathcal {M}_{\lambda , \Lambda }^{+}(D^2 u) = f(x) & \!\text{in} & \!\Omega \\[3pt] u(x) = g(x) & \!\text{on} & \!\partial \Omega , \end{array} \right.} \end{equation*}$$for a bounded, regular, and open set normalΩ⊂Rn$\Omega \subset \mathbb {R}^n$, and appropriate continuous data pfalse(·false),qfalse(·false)$p(\cdot ), q(\cdot )$, ffalse(·false)$f(\cdot )$, and gfalse(·false)$g(\cdot )$. Such sharp regularity estimates generalize and improve, to some extent, earlier ones via geometric treatments. Our results are consequences of geometric tangential methods and make use of compactness, localized oscillating, and scaling techniques. In the end, our findings are applied in the study of a wide class of nonlinear models.