Equivalence of coercivity and mean coercivity in higher-order variational integrals with application to minimization

We consider a functional of the type F(u,Ω)=∫ΩF(Dku(x))dx on the Dirichlet class, where F is a continuous function and Ω is an open bounded set of Rn with a Lipschitz boundary. We prove that coercivity and mean coercivity are equivalent under growth conditions, and further we prove that mean coercivity and quasiconvexity are equivalent. Subsequently, we deduce that F(u,Ω) has a minimum under the condition that the integrand F satisfies the growth condition and mean coercivity.

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