Intrinsically p-biharmonic maps

Reference:

Hornung, P. and Moser, R., 2012. Intrinsically p-biharmonic maps. Submitted paper

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Abstract

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.

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