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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.

In this manuscript, we establish C loc α , α θ regularity estimates for bounded weak solutions of a certain class of doubly degenerate evolution PDEs, whose simplest model case is given by ∂ u ∂ t − d i v ( m | u | m − 1 | ∇ u | p − 2 ∇ u ) = f ( x , t ) in Ω T ≔ Ω × ( 0 , T ) , where m ⩾ 1, p ⩾ 2 and f belongs to a suitable anisotropic Lebesgue space. Employing intrinsic scaling techniques and geometric tangential methods, we derive sharp regularity estimates for such models, which depend only on universal and compatibility parameters of the problem. In this scenario, our results are natural improvements for former ones in the context of nonlinear evolution PDEs with degenerate structure via a unified approach. As a consequence of our findings and approach, we address a Liouville type result for entire weak solutions of a related homogeneous problem with frozen coefficients and asymptotic estimates under a certain approximating regime, which may have their own mathematical interest. We also present examples of degenerate PDEs where our results can be applied.