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**Sobolev spaces and harmonic maps for metric space targets.**
*(English)*
Zbl 0862.58004

This article extends the notion of Sobolev \(W_{1,p}\) spaces to spaces of functions with a complete metric space as target. Based on this new definition the article provides a systematic development of the resulting theory of generalized Sobolev methods together with core applications. The notion of \(L_p\)-integrability can naturally be extended to functions with values in metric spaces. There is no derivative for metric space valued functions but the \(p\)-th power of the “directional difference distance quotient” \(1/t (d(u(x) - u(x+tV))\) gives measures, which are proven to converge (weakly) to the “\(p\)-energy measure” \(e(x)dx\). For \(p\) greater than 1 this measure is Lebesgue absolutely continuous. Based on these results the authors manage to develop many of the important results of Sobolev calculus in this generalized context. These include: Lower continuity of the Sobolev energies, construction of directional energy measures together with various estimates, a differentiation theory for directional energies, a natural precompactness theorem and more. In the second part of the article least energy maps with prescribed values on the boundary and solutions of the Dirichlet problem with values in nonpositive curvature metric spaces are constructed.

The results represent strong generalizations of work on harmonic maps into smooth manifolds of nonpositive curvature by J. Eells and J. H. Sampson [Am. J. Math. 86, 109-160 (1964; Zbl 0122.40102)], R. S. Hamilton [‘Harmonic maps of manifolds with boundary’ (Lect. Notes Math. 471) (1975; Zbl 0308.35003)] and Schoen. The paper is written in a diligent and clear style. While many proofs require sensitive estimates, the geometric nature of the arguments make the architecture of the theory transparent. The substantial results in this paper indicate that this approach is natural even though it is technically involved.

The results represent strong generalizations of work on harmonic maps into smooth manifolds of nonpositive curvature by J. Eells and J. H. Sampson [Am. J. Math. 86, 109-160 (1964; Zbl 0122.40102)], R. S. Hamilton [‘Harmonic maps of manifolds with boundary’ (Lect. Notes Math. 471) (1975; Zbl 0308.35003)] and Schoen. The paper is written in a diligent and clear style. While many proofs require sensitive estimates, the geometric nature of the arguments make the architecture of the theory transparent. The substantial results in this paper indicate that this approach is natural even though it is technically involved.

Reviewer: C.Günther (Libby)

### MSC:

58B10 | Differentiability questions for infinite-dimensional manifolds |

58D15 | Manifolds of mappings |

58E20 | Harmonic maps, etc. |

26E25 | Set-valued functions |

58C20 | Differentiation theory (Gateaux, Fréchet, etc.) on manifolds |

49K27 | Optimality conditions for problems in abstract spaces |