Intrinsically \(p\)-biharmonic maps
In this paper, the authors introduce and study an intrinsic version of the \(p\)-biharmonic energy functional of the form \(E_p(u) = \int_\Omega |Ddu|^p\) for maps \(u: \Omega \to N\). Here \(\Omega\subset {\mathbb R}^m\) is a domain and \(N\) is a compact Riemannian manifold \(N\) without boundary. This functional can be considered as a generalization of functionals for \(p\)-harmonic maps and biharmonic maps. In [\textit{R. Moser}, Commun. Partial Differ. Equations 33, No. 9, 1654--1689 (2008; Zbl 1154.58007)], minimizers of \(E_2\) were constructed in the space \(H^2_N(\Omega)\) and for \(p >2\) it is not difficult to generalize the arguments. Thus, in this paper, the authors mostly focus on the case \(1<p<2\). However if \(p \in (1,2)\) and if \(u \in W^{1,p}(\Omega, {\mathbb R}^n)\), then the Hessian of \(u\) is no longer well-defined as a distribution when we assume \(N\) is embedded into \({\mathbb R}^n\). To overcome this difficulty, the authors introduce a natural intrinsic Sobolev space and prove, by means of the direct method, existence of minimizers of \(E_p\). More precisely, for embedding \(N \subset {\mathbb R}^n\), let \(W^{k,p}(\Omega, N) = \{u\in W^{k,p}(\Omega, {\mathbb R}^n): u(x) \in N\,\, \text{a.e.}\}\), and denote by \(W^{2,p}_N(\Omega)\) the set of all \(u \in W^{1,p}(\Omega, N)\) such that \(\partial_\alpha u \in W^{1,p}(\Omega, \Gamma)\) for all \(\alpha = 1,2, \dots, m\), where \(\Gamma = u^{-1}TN\). For \(u \in W^{2,p}_N(\Omega)\), define \(\| D^\Gamma du\|_{L^p(\Omega)}^p = \sum_{\alpha =1}^n \| D^\Gamma \partial_\alpha u\|_{L^p(\Omega)}^p\) and set \[ E_p(u) = \| D^\Gamma du\|_{L^p(\Omega)}^p. \] The authors prove that if \(p \in (1,2)\) and \(u_0 \in W^{2,p}_N(\Omega)\), then the functional \(E_p\) attains its minimum among all \(u \in W^{2,p}_N(\Omega)\) satisfying \((u, du) = (u_0, du_0)\) on \(\partial \Omega\) in the sense of traces. They also derive a monotonicity formula similar to a well-known formula for harmonic maps [\textit{P. Price}, Manuscr. Math. 43, 131--166 (1983; Zbl 0521.58024)], which plays an important role in the regularity theories for the problems in supercritical dimensions. Finally, the authors consider more general functionals defined in terms of convex or polyconvex functions \(f\) of the form \(\int_\Omega f(Ddu)\).
scientific article; zbMATH DE number 6369369
scientific article