For p-Laplace type equations, Phi(Du) is C^k whenever Phi is positively homogeneous of degree m with m>k/alpha and Du is C^{0,alpha}.
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Higher Regularity of Homogeneous Gradient Compositions for $p$-Laplace-Type Equations
For p-Laplace type equations, Phi(Du) is C^k whenever Phi is positively homogeneous of degree m with m>k/alpha and Du is C^{0,alpha}.