For a three-dimensional space, an isotropic tensor-valued function Φ of a symmetric tensor A has the representation Φ(A) = α(A)I + β(A)A + γ(A)A2, where the coefficients α, β, γ are isotropic scalar-valued functions. It is known that these coefficients may fail to be as smooth as Φ at those tensors A that do not have three distinct eigenvalues. Serrin and Man determined conditions on the smoothness of Φ that guarantee the existence of continuous coefficients. We give a different proof of their results and also determine conditions on Φ that guarantee the existence of continuously differentiable coefficients.
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