Coercivity of integrals versus coercivity of integrands
Considering the integral functional of the calculus of variations \[ I(u)=\int_{\Omega} f(x,u(x),\nabla u(x)) dx \] where \(\Omega\subset \mathbb{R}^n, u:\Omega \to \mathbb{R},\) the authors discuss the possibility that a coercivity assumption on \(I\) implies an analogous property on the integrand. More precisely, they consider the conditions \[ \begin{alignedat}{2} I(u)&\geq \alpha |\nabla u|^p_{L^p}+\beta, \qquad\qquad &&u\in W^{1,\infty}_0(\Omega) \quad [ u\in W^{1,\infty}(\Omega) ] \tag{1}\\ f(x,u,\xi) &\geq a|\xi|^p+b, &&(x,u,\xi)\in \Omega\times\mathbb{R}\times \mathbb{R}^n.\tag{2}\end{alignedat} \] They prove that (1) and (2) are equivalent if \(f=f(\xi)\) but the implication \((1)\Longrightarrow (2)\) is false for integrands of the type \(f(x,\xi)\) or \(f(u,\xi).\) They also extend these results to the vectorial case, i.e., for \(u:\Omega\to \mathbb{R}^m\), \(m>1\).
scientific article; zbMATH DE number 765087