Lee, Yong Hah Uniqueness of the boundary value problem of harmonic maps via harmonic boundary. (English) Zbl 1443.58011 Bull. Malays. Math. Sci. Soc. (2) 43, No. 3, 2733-2743 (2020). The author proves the uniqueness of solutions for the boundary value problem of harmonic maps via harmonic boundary. Let \(\mathcal {BD}(M)\) be the set of all bounded continuous functions on a complete Riemannian manifold \(M\) whose distributional gradient \(\nabla f\) belongs to \(L^2(M)\). There exists a locally compact Hausdorff space \(\hat M\), called the Royden compactification of \(M\), which contains \(M\) as an open dense subset [M. Nakai and L. Sario, Classification theory of Riemann surfaces. Berlin etc.: Springer Verlag (1970; Zbl 0199.40603)]. And every function \(f\) in \(\mathcal {BD}(M)\) can be extended to a continuous function on \(\hat M\). The Royden boundary \(\partial \hat M\) is the set \(\hat M \setminus M\) and the harmonic boundary \(\Delta_M\) is defined by \[ \Delta_M = \{x\in \partial \hat M: f(x) = 0\,\,\, \mbox{for all $f\in \mathcal{BD}_0(M)$}\}, \] where \(\mathcal {BD}_0(M)\) is the closure of the set of all compactly supported smooth functions in \(\mathcal {BD}(M)\).Let \(M\) be a complete Riemannian manifold and \(\Delta_M\) be the harmonic boundary of \(M\). Let \(N\) be a complete Riemannian manifold and let \(B_r(p)\) be a geodesic ball in \(N\) which is disjoint from the cut locus of \(p\). The author proves that, for any \(f \in C(\Delta_M, B_r(p))\), there exists a unique harmonic map \(u \in C(M, B_r(p))\), which is a limit of a sequence of harmonic maps with finite total energy in the sense of the supremum norm, such that for each \(x \in \Delta_M\), \[ \lim_{y\in M\to x} u(y) = f(x). \] The existence of solutions for the above boundary value problem of harmonic maps is proven in [Y. H. Lee, Potential Anal. 41, No. 2, 463–468 (2014; Zbl 1300.58006)]. Reviewer: Gabjin Yun (Yongin) MSC: 58E20 Harmonic maps, etc. 53C43 Differential geometric aspects of harmonic maps Keywords:harmonic map; harmonic boundary; boundary value problem; energy Citations:Zbl 0199.40603; Zbl 1300.58006 PDFBibTeX XMLCite \textit{Y. H. Lee}, Bull. Malays. Math. Sci. Soc. (2) 43, No. 3, 2733--2743 (2020; Zbl 1443.58011) Full Text: DOI References: [1] Avilés, P.; Choi, HI; Micallef, M., Boundary behavior of harmonic maps on non-smooth domains and complete negatively curved manifolds, J. Funct. Anal., 99, 293-331 (1991) · Zbl 0805.53037 [2] Giaquinta, M.; Hildebrandt, S., A priori estimates for harmonic mappings, J. Reine Angew. Math., 336, 124-164 (1982) · Zbl 0508.58015 [3] Hildebrandt, S.; Kaul, H.; Widman, K., An existence theorem for harmonic mappings of Riemannian manifolds, Acta Math., 138, 1-16 (1977) · Zbl 0356.53015 [4] Kendall, WS, Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence, Proc. Lond. Math. Soc., 61, 371-406 (1990) · Zbl 0675.58042 [5] Lee, YH, Rough isometry and energy finite solutions of elliptic equations on Riemannian manifolds, Math. Ann., 318, 181-204 (2000) · Zbl 0968.58018 [6] Lee, YH, Asymptotic boundary value problem of harmonic maps via harmonic boundary, Potential Anal., 41, 463-468 (2014) · Zbl 1300.58006 [7] Lee, YH, Royden decomposition for harmonic maps with finite total energy, Results Math., 71, 687-692 (2017) · Zbl 1377.58010 [8] Sario, L.; Nakai, M., Classification Theory of Riemann Surfaces (1970), Berlin: Springer, Berlin · Zbl 0199.40603 [9] Sung, CJ; Tam, LF; Wang, J., Bounded harmonic maps on a class of manifolds, Proc. Am. Math. Soc., 124, 2241-2248 (1996) · Zbl 0865.58015 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.