Intrinsically p-biharmonic maps

For a compact Riemannian manifold $N$, a domain $\Omega \subset \mathbb{R}^m$ and for $p \in (1,\infty)$, we introduce an intrinsic version $E_p$ of the $p$-biharmonic energy functional for maps $u : \Omega \to N$. This requires finding a definition for the intrinsic Hessian of maps u : \Omega \to N$ whose first derivatives are merely $p$-integrable. We prove, by means of the direct method, existence of minimizers of $E_p$ within the corresponding intrinsic Sobolev space, and we derive a monotonicity formula. Finally, we also consider more general functionals defined in terms of polyconvex functions.