Some properties of the Hessian Matrix of a Strictly Convex Function.

A strictly convex function of n real variables is any continuously differentiable funetion with the property (1) below. Strictly convex functions arise naturally in a number of physical theories, notably, in thermodynamics and elasticity theory. It is with these applications in mind that we have been led to consider some properties of the Hessian matrix of a twice continuously differentiable strictly convex function. To motivate and set the theme of the purely mathematical considerations to follow, allow us first to describe briefly two specific physical problems where the theorems we shall prove might be useful. Our first example is Gibbs' theory of the stable equilibrium of a heterogeneous substance. Here one is led to consider an energy function W ( , v, c1? c2, . ..,c^), where is the entropy density, v is the specific volume, and c1? c2 , . . ., CN are the concentrations of N dissolved components of the mixture. According to the theory [l, 2], every homogeneous state of the fluid mixture will be in stable equilibrium with a certain environment if W is a strictly convex function. Our second example is taken from the theory of large elastic deformations. In the classical linear theory of infinitesimal elastic deformations, it is generally assumed on physical grounds that the quadratic energy function of the six infinitesimal strain components is positive definite. Coleman and Noll [3] have found that a natural extension of this condition which applies to the energy function of the theory of large elastic deformations is a type of convexity restriction. Both in the theory of heterogeneous fluids and the theory of large elastic deformations, it is desirable to replace the convexity condition, which is global in nature, by a set of inequalities, local in nature, to be satisfied by the Hessian matrix of the relevant convex function. For example, certain familiär inequalities [4, 2] of thermodynamics are Statements derived from the condition that the Hessian matrix of the internal energy or free energy function be positive definite. But from Theorem I and the remarks preceding Theorem VI, one sees that strict convexity of a function implies only that its Hessian matrix be positive sewi-definite everywhere. Thus if the physical law demands only strict convexity of the energy function, then there may be states of the material where the Hessian matrix has a zero eigenvalue. Such states of a substance should be of particular concern. Thus it is of interest to determine the possible frequency and distribution of such exceptional states. It is primarily to this question that we address the following remarks. Let #*, # , . . . , x ränge over the rectilinear coordinates of some domain in affine ra-dimensional space, 5l. By a domain we shall mean an open set plus perhaps some or all of its boundary.