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Finding surface mappings with least distortion arises from many applications in various fields. Extremal Teichmüller maps are surface mappings with least conformality distortion. The existence and uniqueness of the extremal Teichmüller map between Riemann surfaces of finite type are theoretically guaranteed (see Fletcher and Markovic, Quasiconformal maps and Teichmüller theory, Oxford Graduate Texts in Math., vol. 11, Oxford University Press, Oxford, 2007). Recently, a simple iterative algorithm for computing the Teichmüller maps between connected Riemann surfaces with given boundary value was proposed by Lui, Lam, Yau, and Gu in Teichmüller extremal mapping and its applications to landmark matching registration, arXiv:1211.2569. Numerical results were reported in the paper to show the effectiveness of the algorithm. The method was successfully applied to landmarkmatching registration. The purpose of this paper is to prove the iterative algorithm proposed in loc. cit., indeed converges.

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This paper concerns the well-posedness theory of the motion of a physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in (Coutand et al., Commun Math Phys 296:559–587, 2010; Coutand and Shkoller, Arch Ration Mech Anal 206:515–616, 2012; Jang andMasmoudi,Well-posedness of compressible Euler equations in a physical vacuum, 2008) by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary.

In this paper, we prove the Lipschitz regularity of continuous harmonic maps from an finite dimensional Alexandrov space to a compact smooth Riemannian manifold. This solves a conjecture of F. H. Lin in [36]. The proof extends the argument of Huang-Wang [26].