Lee, Yong Hah Asymptotic boundary value problem of harmonic maps via harmonic boundary. (English) Zbl 1300.58006 Potential Anal. 41, No. 2, 463-468 (2014). In this paper, the author proves the existence of harmonic map between complete Riemannian manifolds taking the given boundary data on the harmonic boundary of the domain manifold. Let \(M\) be a complete Riemannian manifold and let \(\widehat M\) be its Royden compactification, i.e., \(\widehat M\) is a locally compact Hausdorff space which contains \(M\) as an open dense subset. Then the harmonic boundary \(\Delta_M\) of \(M\) is defined by \[ \Delta_M = \{\mathbf{x}\in \partial \widehat M\,:\, f(\text\textbf{x}) = 0 \,\, \text{for all \(f \in {\mathcal {BD}_0}(M)\)}\}, \] where \(\partial \widehat M = \widehat M\setminus M\) is the Royden boundary and \({\mathcal {BD}_0}(M)\) is the closure of the set of all compactly supported smooth functions in the space of all bounded continuous functions on \(M\) whose distributional gradient \(\nabla f\) belongs to \(L^2(M)\). Let \(B_r(p)\) be a geodesic ball in another complete Riemannian manifold \(N\) which lies within the normal range of \(p\) in \(N\). The author shows that if \(f \in C(\Delta_M, B_r(p))\), there exists a harmonic map \(u \in C(M, B_r(p))\) such that for each \(\mathbf{x}\in \Delta_M\), \[ \lim_{x\in M \to \mathbf{x}} u(x) = f(\mathbf{x}). \] Reviewer: Gabjin Yun (Yongin) Cited in 1 ReviewCited in 1 Document MSC: 58E20 Harmonic maps, etc. 53C43 Differential geometric aspects of harmonic maps Keywords:harmonic map; harmonic boundary; asymptotic boundary value problem PDFBibTeX XMLCite \textit{Y. H. Lee}, Potential Anal. 41, No. 2, 463--468 (2014; Zbl 1300.58006) Full Text: DOI References: [1] Avilés, P., Choi, H.I., 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.: An existence theorem for harmonic mappings of Riemannian manifolds. J. Reine Angew. Math. 336, 124-164 (1982) [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] Sario, L., Nakai, M.: Classification Theory of Riemann Surfaces. Springer Verlag, Berlin, Heidelberg, New York (1970) · Zbl 0199.40603 [5] Sung, C.J., Tam, L.F., Wang, J.: Bounded harmonic maps on a class of manifolds. Proc. Amer. 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.