In this paper, we prove borderline gradient continuity of viscosity solutions to Fully nonlinear elliptic equations at the boundary of a $C^{1,\dini}$-domain. Our main result Theorem 3.1 is a sharpening of the boundary gradient estimate proved by Ma-Wang following the borderline interior gradient regularity estimates established Daskalopoulos-Kuusi-Mingione. We however mention that, differently from the approach in the interior case which depends on $W^{1,q}$ estimates, our proof is slightly more geometric and is based on compactness arguments inspired by the techniques in the fundamental works of Caffarelli.

中文翻译：

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Dini 域中完全非线性方程的边界正则性

在本文中，我们证明了 $C^{1,\dini}$ 域边界处完全非线性椭圆方程的粘度解的边界梯度连续性。我们的主要结果定理 3.1 是根据 Daskalopoulos-Kuusi-Mingione 建立的边界内部梯度规律估计，对 Ma-Wang 证明的边界梯度估计进行了锐化。然而，我们提到，与依赖于 $W^{1,q}$ 估计的内部情况中的方法不同，我们的证明稍微更几何，并且基于受 Caffarelli 基本工作中的技术启发的紧凑性论证。