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The Vectorial Case

  • Bernard Dacorogna
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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Bernard Dacorogna
    • 1
  1. 1.Département de MathématiquesÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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