Harmonic maps between Alexandrov spaces
type: Item
instance of: scholarly article
title: Harmonic maps between Alexandrov spaces
zbMATH Open document ID: 1376.58004
author: Jia-Cheng Huang
author: Hui-Chun Zhang
published in: The Journal of Geometric Analysis
publication date: 2017-08-03T00:00:00Z
review text: In this paper, the authors study the existence, uniqueness and regularity of harmonic maps from an Alexandrov space into a geodesic space with curvature \(\leq 1\) in the sense of Alexandrov. Let \(X\) be an Alexandrov space of Hausdorff dimension \(n\) with curvature \(\geq k\) and an \(n\)-dimensional Hausdorff measure \(\mu\), \(\Omega \subset X\) be a bounded domain, and let \(Y\) be a complete geodesic space with curvature \(\leq 1\) in the sense of Alexandrov. Given \(\epsilon >0\) and a Borel measurable function \(u: \Omega \to Y\), the approximating energy functional \(E^u_\epsilon\) of \(u\) is defined as follows: For each compactly supported continuous function \(\varphi \in C_c(\Omega)\), \[ E^u_\epsilon(\varphi) := C_n \int_\Omega \varphi(x) d \mu(x) \int_{B_\epsilon(x)\cap \Omega} \frac{d^2(u(x), u(y))}{\epsilon^{n+2}}d \mu(y), \] where \(C_n\) is a normalized constant. And the energy functional of \(u\) is defined by \[ E^u(\varphi) = \limsup_{\epsilon \to 0} E^u_{\epsilon}(\varphi),\quad \forall \varphi \in C_c(\Omega). \] We say that \(u \in W^{1, 2}(\Omega, Y)\) if \(u \in L^2(\Omega)\) and it has finite energy \[ \sup_{\varphi\in C_c(\Omega), 0 \leq \varphi \leq 1} E^u(\varphi) <\infty. \] For a ball \(B_\rho(q) \subset Y\) with \(\rho< \pi/2\) and \(\varphi \in W^{1,2}(\Omega, Y)\) with \(\varphi(\Omega) \subset B_\rho(q)\), let \[ W^{1,2}_\varphi (\Omega, B_\rho(q)):= \{v \in W^{1,2}(\Omega, Y)\,:\, d(v, \varphi) \in W^{1,2}_0, v(\Omega) \subset B_\rho(q)\,\}. \] The first result proved by authors in this paper is that \(W^{1,2}_\varphi (\Omega, B_\rho(q))\) has a unique element \(u\) of least energy. It is called the harmonic map on \(\Omega\) which agrees \(\varphi\) on \(\partial \Omega\). The second result is the regularity of harmonic maps given in the above. Assume both \(\Omega\) and \(X\setminus \overline{\Omega}\) satisfy the measure density condition, i.e., there exists a constant \(C >0\) such that \(\mu(B_r(x) \cap \Omega) \geq C \mu(…
reviewed by: Gabjin Yun
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C43
zbMATH DE Number: 6756126
zbMATH Keywords: harmonic maps
zbMATH Keywords: Alexandrov spaces
zbMATH Keywords: geodesic space
zbMATH Keywords: energy on Alexandrov spaces
zbMATH Keywords: positive upper curvature bounds
zbMATH Keywords: global regularity
MaRDI profile type: Publication
full work available at URL: s12220-016-9722-y
OpenAlex ID: W2471259535
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DOI: 10.1007/S12220-016-9722-Y
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description: scientific article; zbMATH DE number 6756126