We establish sharp \(C_{\text {loc}}^{1, \beta }\) geometric regularity estimates for bounded solutions of a class of fully nonlinear elliptic equations with non-homogeneous degeneracy, whose model equation is given by

for a bounded and open set \(\Omega \subset {\mathbb {R}}^N\), and appropriate data \(p, q \in (0, \infty )\), \(\mathfrak {a}\) and f. Such regularity estimates simplify and generalize, to some extent, earlier ones via totally different modus operandi. Our approach is based on geometric tangential methods and makes use of a refined oscillation mechanism combined with compactness and scaling techniques. In the end, we present some connections of our findings with a variety of nonlinear geometric free boundary problems and relevant nonlinear models in the theory of elliptic PDEs, which may have their own interest. We also deliver explicit examples where our results are sharp.

我们为一类具有非齐次退化的完全非线性椭圆方程的有界解建立了尖锐的\（C _ {\ text {loc}} ^ {1，\ beta} \）几何正则性估计，其方程由下式给出：

对于有界和开放集\（\ Omega \ subset {\ mathbb {R}} ^ N \）和适当的数据\（p，q \ in（0，\ infty）\），\（\ mathfrak {a} \）和f。这样的规律性估计通过完全不同的方式操作在某种程度上简化和概括了早期的规律性。我们的方法基于几何切向方法，并结合了紧凑性和缩放技术，使用了完善的振荡机制。最后，我们介绍了我们的发现与椭圆PDE理论中的各种非线性几何自由边界问题和相关非线性模型之间的一些联系，这可能有其自己的兴趣。我们还提供了一些明确的示例，这些示例中我们的结果非常清晰。