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Uniqueness of the boundary value problem of harmonic maps via harmonic boundary. (English) Zbl 1443.58011

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)].

MSC:

58E20 Harmonic maps, etc.
53C43 Differential geometric aspects of harmonic maps
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