Non-Convex Integrands

Part of the Applied Mathematical Sciences book series (AMS, volume 78)


In Chapter 3 and 4 we have seen that in order to get existence theorems for
$$ \inf {\mkern 1mu} \{ {\mkern 1mu} I(u){\mkern 1mu} = {\mkern 1mu} \int\limits_\Omega {f(x,{\mkern 1mu} u(x),{\mkern 1mu} \nabla u(x)){\mkern 1mu} dx{\mkern 1mu} :{\mkern 1mu} u{\mkern 1mu} \in {\mkern 1mu} {u_0}{\mkern 1mu} + {\mkern 1mu} W_0^{1,p}(\Omega ;{\mathbb{R}^m})} \}$$
he convexity (or quasiconvexity in the vectorial case) of f, with respect to the last variable, plays a central role. In this chapter we shall study the case where f fails to be convex (quasiconvex in the vectorial case).


Minimal Surface Existence Theorem Lower Semicontinuous Scalar Case Reverse Inequality 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  1. 1.Département de MathématiquesÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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