Integral convexity and parabolic systems

In this work we give optimal, i.e. necessary and sufficient, conditions for integrals of the Calculus of Variations to guarantee the existence of solutions – both weak and variational solutions – to the associated L2-gradient flow. The initial values are merely L^2-functions with possibly infinite energy. In this context, the notion of integral convexity, i.e. the convexity of the variational integral and not of the integrand, plays the crucial role; surprisingly, this type of convexity is weaker than the convexity of the integrand. We demonstrate this by means of certain quasi-convex and non-convex integrands.