# The Vectorial Case

• Bernard Dacorogna
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 78)

## Abstract

We now turn our attention to the vectorial case. Recall that
$$I(u){\mkern 1mu} = {\mkern 1mu} \int\limits_\Omega {f(x,{\mkern 1mu} u(x),{\mkern 1mu} \nabla u(x)){\mkern 1mu} dx}$$
(1)
and u : Ω ⊂ ℝn → ℝm (thus ▽u ∈ ℝnm),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 W1,p(Ω, ℝm), 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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