Direct Methods in the Calculus of Variations pp 97-195 | Cite as

# The Vectorial Case

Chapter

## Abstract

We now turn our attention to the vectorial case. Recall that
and

$$
I(u){\mkern 1mu} = {\mkern 1mu} \int\limits_\Omega {f(x,{\mkern 1mu} u(x),{\mkern 1mu} \nabla u(x)){\mkern 1mu} dx}$$

(1)

*u*: Ω ⊂ ℝ^{n}→ ℝ^{m}(thus ▽*u*∈ ℝ^{nm}),with*n*,*m*> 1. While the convexity of*f*with respect to the last variable ▽*u*is playing the central role in the scalar case (*m*= 1 or*n*= 1), cf. Chapter 3., and is still sufficient, in the vectorial case, to ensure weak lower semicontinuity of*I*in W^{1,p}(Ω, ℝ^{m}), it is far from being a necessary condition. Such a condition is the so-called*quasiconvexity*introduced by Morrey. However it is hard to verify, in practice, if a given function*f*is quasiconvex, since it is not pointwise condition. Therefore one is lead to introduce a slightly weaker condition known as*rank one convexity*and a stronger condition, introduced by Ball, called*polyconvexity*. One can relate all these definitions through the following diagram (Fig. 4.1).## Keywords

Existence Theorem Elementary Property Scalar Case Coercivity Condition Weak Continuity
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© Springer-Verlag Berlin Heidelberg 1989