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Bent Fuglede
2018 Transactions of the American Mathematical Society  
This is a continuation of the Cambridge Tract "Harmonic maps between Riemannian polyhedra", by J. Eells and the present author. The variational solution to the Dirichlet problem for harmonic maps with countinuous boundary data is shown to be continuous up to the boundary, and thereby uniquely determined. The domain space is a compact admissible Riemannian polyhedron with boundary, while the target can be, for example, a simply connected complete geodesic space of nonpositive Alexandrov
more » ... ; alternatively, the target may have upper bounded curvature provided that the maps have a suitably small range. Essentially in the former setting it is further shown that a harmonic map pulls convex functions in the target back to subharmonic functions in the domain. 757 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 758 BENT FUGLEDE theorem we establish uniqueness of a harmonic map with prescribed continuous restriction to the boundary. This is done by proving the same maximum principle for the distance function d Y (φ 0 , φ 1 ) between two harmonic maps φ 0 , φ 1 which was obtained by Jäger and Kaul [JK] in the case where X and Y are Riemannian manifolds. Also in our setting, d Y (φ 0 , φ 1 ) is therefore subharmonic in case Y has nonpositive curvature (Theorem 1 (b)), as shown by considering the map φ 0 ×φ 1 : X → Y ×Y . In the case of upper bounded curvature (Theorem 3 (b)), d Y (φ 0 , φ 1 ) is expressed in terms of a subsolution to the very same elliptic operator of divergence type as devised in [JK]. Summing up, the Dirichlet problem is uniquely solvable and well posed. Similar results are obtained for the Dirichlet problem for maps into a smooth Riemannian manifold without boundary (Theorems 4 and 5). The study of maps into a manifold is not quite a particular case of that of a geodesic space target because our concept of energy of maps into a geodesic space, extending that of Korevaar and Schoen [KS] for maps from a (smooth) Riemannian manifold, requires that the Riemannian metric g on the domain polyhedron X be simplexwise smooth, while bounded measurable components of g suffice for a good concept of energy of maps into a Riemannian manifold. When g is simplexwise smooth, the two concepts of energy are identical (up to a dimensional constant factor) provided either that φ(X) is bounded [EF, Theorem 9.2] or that the target manifold is complete and simply connected with nonpositive sectional curvature (Proposition 2 below). Recall from [EF, Chapter 4] that a polyhedron X is termed admissible if it is dimensionally homogeneous, say of dimension m, and if (in case m ≥ 2) any two m-simplexes of X with a common face σ (dim σ = 0, 1, . . . , m − 2) can be joined by a chain of m-simplexes containing σ, any two consecutive ones of which have a common (m − 1)-face containing σ. We denote throughout by X an m-dimensional admissible polyhedron, connected and locally finite, and endowed with a nondegenerate Riemannian metric g whose restriction to each open m-simplex of X is at least bounded and measurable. The associated volume measure on X is denoted by µ g = µ, the intrinsic (Riemannian) distance on X by d g X = d X , and the closed ball with center x ∈ X and radius r by B X (x, r). Based on the work of Korevaar and Schoen [KS] a concept of energy of a map φ of (X, g) into a metric space (Y, d Y ) is developed in [EF, Chapter 9], assuming that g is simplexwise smooth, i.e., g shall be smooth in every open m-simplex s of X, and g |s shall extend smoothly to a (nondegenerate) Riemannian metric on the affine m-space containing s [EF, Remark 4.
doi:10.1090/s0002-9947-04-03498-1 fatcat:rpteoco2czebvfizy7tpzijh5a