Smoothness of the Scalar Coefficients in the Representations of Isotropic Tensor-Valued Functions

摘要

For a three-dimensional space, an isotropic tensor-valued function omega of asymmetric tensor A has the representation omega(A) = alpha(A)I + beta(A)A + gamma(A)A2, in which the coefficients alpha, beta, and gamma are isotropic scalar-valued functions. It is known that these coefficients may fail to be as smooth as omega at those tensors A that do not have three distinct eigenvalues. Serrin (1959) and Man (1994) determined conditions on the smoothness of P that guarantee the existence of continuous coefficients. We give a different proof of their results and also determine conditions on P that guarantee the existence of continuously differentiable coefficients. (AN).