The Calculus of Variations under Convexity Assumptions

Abstract

The central focus of the calculus of variations is the functional

$$ I\left( u \right) = \int {_\Omega\phi \left( {x,u(x),\nabla u(x)} \right)dx} $$

where the integrand ϕ explicitly depends upon the gradient variable ∇ u.Ωis assumed to be an open, regular, bounded domain of R ^N.The admissible functions u:Ω → R ^m belong to some reflexive Sobolev space and they may satisfy some other restriction like having the boundary values prescribed.The integrand ϕ:Ω × R ^m × M ^{m × N} → R * is assumed to be a Carathéodory function. By this we simply mean that φ is measurable on thexvariable and continuous with respect to u and ∇ u. We may eventually let ϕ take on the value +∞ as indicated by R*=R∪ { +∞ }.