Regularity of solutions to degenerate normalized $p$-Laplacian equation with general variable exponents

In this paper, we consider a kind of degenerate normalized $p$-Laplacian equation with general variable exponents. We establish local $C^{1,\alpha'}$ regularity of viscosity solutions by making use of the compactness argument, scaling techniques and the localized oscillating method. In addition, we also obtain almost optimal pointwise $C^{1,\tau}$ regularity for degenerate free transmission problem related to normalized $p$-Laplacian. Our argument is based on a new improved oscillation-type estimate combined with a localized analysis.