SMOOTHNESS OF THE SCALAR COEFFICIENTS IN REPRESENTATIONS OF ISOTROPICTENSER-VALUED FUNCTIONS

For a three-dimensional space, an isotropic tenser-valued function Phi of a symmetric tensor A has the representation Phi(A) = alpha(A)I + b eta(A)A + gamma(A)A(2), where the coefficients alpha, beta, gamma are isotropic sealar-valued functions. It is known that these coefficients may fail to be as smooth as Phi at those tensors A that do not have t hree distinct eigenvalues. Serrin and Man determined conditions on the smoothness of Phi that guarantee the existence of continuous coeffici ents. We give a different proof of their results and also determine co nditions on Phi that guarantee the existence of continuously different iable coefficients.