Chang, Sun-Yung A.; Wang, Lihe; Yang, Paul C. A regularity theory of biharmonic maps. (English) Zbl 0953.58013 Commun. Pure Appl. Math. 52, No. 9, 1113-1137 (1999). The authors consider mappings \(u:\Omega\to S^k=\{z\in \mathbb{R}^{k +1}: |z|=1\}\) from an \(n\)-dimensional domain \(\Omega\) into the unit \(k\)-sphere which are of Sobolev class \(W^2_2\), and in addition critical points of the energy \(\int_\Omega\sum^{k+1}_{\alpha=1} (\Delta_gu^\alpha)^2dV_g\) where \(\Delta_g\) and \(dV_g\) denote the Laplacian and the volume element w.r.t. the given metric \(g\) on \(\Omega\).The following regularity results are established for this class of biharmonic mapping: any biharmonic map in \(W^2_2\) defined on a four-disk is Hölder continuous. If in addition a biharmonic map defined on a \(n\)-dimensional domain \((n\geq 5)\) is also stationary, then the \((n-4)\)-dimensional Hausdorff measure of the singular set is zero. Moreover, any continuous biharmonic map is smooth. Reviewer: M.Fuchs (Saarbrücken) Cited in 2 ReviewsCited in 69 Documents MSC: 58E20 Harmonic maps, etc. 31C05 Harmonic, subharmonic, superharmonic functions on other spaces 58E35 Variational inequalities (global problems) in infinite-dimensional spaces Keywords:regularity; biharmonic mapping PDFBibTeX XMLCite \textit{S.-Y. A. Chang} et al., Commun. Pure Appl. Math. 52, No. 9, 1113--1137 (1999; Zbl 0953.58013) Full Text: DOI References: [1] Chang, Amer J Math [2] Chang, Comm Pure Appl Math 52 pp 1099– (1999) · Zbl 1044.58019 · doi:10.1002/(SICI)1097-0312(199909)52:9<1099::AID-CPA3>3.0.CO;2-O [3] Evans, Arch Rational Mech Anal 116 pp 101– (1991) · Zbl 0754.58007 · doi:10.1007/BF00375587 [4] Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies, 105. Princeton University Press, Princeton, N.J., 1983. [5] Hardt, Comm Pure Appl Math 40 pp 555– (1987) · Zbl 0646.49007 · doi:10.1002/cpa.3160400503 [6] ; Private communication. · Zbl 1015.91507 [7] Hélein, C R Acad Sci Paris Sér I Math 312 pp 591– (1991) [8] John, Comm Pure Appl Math 14 pp 415– (1961) · Zbl 0102.04302 · doi:10.1002/cpa.3160140317 [9] Lewis, Proc Amer Math Soc 80 pp 259– (1980) · doi:10.1090/S0002-9939-1980-0577755-6 [10] Nirenberg, Ann Scuola Norm Sup Pisa (3) 13 pp 115– (1959) [11] Uhlenbeck, Acta Math 138 pp 219– (1977) · Zbl 0372.35030 · doi:10.1007/BF02392316 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.