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In this paper, we show the Yau’s gradient estimate for harmonic maps into a metric space (X, dX) with curvature bounded above by a constant κ (κ ⩾ 0) in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of Cheng (1980) and Choi (1982) to harmonic maps into singular spaces.
Let σ > 1 and let M be a complete Riemannian manifold. In a very recent work (Grigor′yan and Sun 2014), Grigor ′ yan and Sun proved that a Liouville type theorem holds for nonnegative solutions of elliptic inequality * Δ u ( x ) + u σ ( x ) ⩽ 0 , x ∈ M . $$ {\Delta} u(x)+u^{\sigma}(x)\leqslant 0,\qquad x\in M. $$ via a pointwise condition...
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