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We show that viscosity solutions to the normalized $p(x)$-Laplace equationcoincide with distributional weak solutions to the strong $p(x)$-Laplaceequation when $p$ is Lipschitz and $\inf p>1$. This yields $C^{1,\alpha}$regularity for the viscosity solutions of the normalized $p(x)$-Laplaceequation. As an additional application, we prove a Rad\'o-type removabilitytheorem.

Equivalence of viscosity and weak solutions for the normalized $p(x)$-Laplacian