Intrinsically \(p\)-biharmonic maps
type: Item
instance of: scholarly article
title: Intrinsically \(p\)-biharmonic maps
zbMATH Open document ID: 1314.58006
author: Peter Hornung
author: Roger Moser
published in: Calculus of Variations and Partial Differential Equations
publication date: 2014-11-13T00:00:00Z
review text: 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…
reviewed by: Gabjin Yun
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 35J35
zbMATH DE Number: 6369369
zbMATH Keywords: biharmonic map
zbMATH Keywords: harmonic map
zbMATH Keywords: p-harmonic map
zbMATH Keywords: monotonicity formula
MaRDI profile type: Publication
full work available at URL: s00526-013-0688-3
OpenAlex ID: W2000948742
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DOI: 10.1007/S00526-013-0688-3
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description: scientific article; zbMATH DE number 6369369