We examine a transmission problem driven by a degenerate quasilinear operator with a natural interface condition. Two aspects of the problem entail genuine difficulties in the analysis: the absence of representation formulas for the operator and the degenerate nature of the diffusion process. Our arguments circumvent these difficulties and lead to new regularity estimates. For bounded interface data, we prove the local boundedness of weak solutions and establish an estimate for their gradient in \(\textrm{BMO}\)-spaces. The latter implies solutions are of class \(C^{0,\mathrm{Log-Lip}}\) across the interface. Relaxing the assumptions on the data, we establish local Hölder continuity for the solutions.

我们研究由具有自然界面条件的简并拟线性算子驱动的传输问题。该问题的两个方面给分析带来了真正的困难：缺乏算子的表示公式以及扩散过程的退化性质。我们的论点规避了这些困难并导致了新的规律性估计。对于有界界面数据，我们证明了弱解的局部有界性，并对其在\(\textrm{BMO}\)空间中的梯度进行了估计。后者意味着解决方案是跨接口的类\(C^{0,\mathrm{Log-Lip}}\) 。放宽对数据的假设，我们为解建立局部霍尔德连续性。