ABSTRACT
Like finite-dimensional minimization problems, the framework for studying the typical problem has two vital components: compactness and lower semicontinuity (in some topological sense). The nonlinear constraint det F > 0 makes convexity of W(x, ·) totally unrealistic in more than one space dimension as, among many other reasons, it contradicts the principle of frame indifference. This chapter presents an existence result that extends to the nonhomogeneous case that of Ball and, like the results of Ball, is partly motivated by the work of Ogden. In addition, the discussion of the regularity of weak solutions represents a novel approach that does not require the consideration of the delicate phase plane analysis. The chapter describes the physical significance of the homogeneity property used for the discussion of regularity.
