A relaxation of the intrinsic biharmonic energy (Q444183)

Given a bounded domain \(\Omega\subset\mathbb R^3\) or \(\mathbb R^4\), and \(N\subset\mathbb R^n\) a compact Riemannian manifold, the author studies the biharmonic energy \[ T(u)= \int_\Omega |\tau(u)|^2,\quad u\in H^1(\Omega,N), \] where \(\tau(u)\in L^2(\Omega,\mathbb R^n)\) denotes the tensor field of \(u\). As shown earlier, a natural space for \(T\) is \[ K(\Omega, N)= \{u\in H^1(\Omega,N)\mid\tau(u)\in L^2(\Omega, \mathbb{R}^n),\,\text{div\,}S(u)+ \tau(u)\cdot\nabla u= 0\}, \] where \(S\) is the stress-energy. The purpose of this paper is to make a framework for the direct method of calculus of variations for \(T\); in particular, to construct minimizers and study the relation to the critical points of \(T\).