Sobolev Mappings between Manifolds and Metric Spaces

Hajłasz, Piotr

doi:10.1007/978-0-387-85648-3_7

Cited by 36 publications

(25 citation statements)

References 78 publications

(100 reference statements)

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“…First, for the energies J and domains Ω of our interest we consider variational problem formulations in W 1,2 (Ω, M ) and in general this space does not possess the structure of a Banach manifold [16,26,27]. Hence the results based on Banach manifolds cannot be used.…”

Section: Céa's Lemma Using Geodesic Homotopiesmentioning

confidence: 99%

Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements

Hardering¹,

Sander²

2014

Found Comput Math

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We prove optimal bounds for the discretization error of geodesic finite elements for variational partial differential equations for functions that map into a nonlinear space. For this we first generalize the well-known Céa lemma to nonlinear function spaces. In a second step we prove optimal interpolation error estimates for pointwise interpolation by geodesic finite elements of arbitrary order. These two results are both of independent interest. Together they yield optimal a priori error estimates for a large class of manifold-valued variational problems. We measure the discretization error both intrinsically using an H 1 -type Finsler norm, and with the H 1 -norm using embeddings of the codomain in a linear space. To measure the regularity of the solution we propose a nonstandard smoothness descriptor for manifold-valued functions, which bounds additional terms not captured by Sobolev norms. As an application we obtain optimal a priori error estimates for discretizations of smooth harmonic maps using geodesic finite elements, yielding the first high order scheme for this problem.

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Section: Céa's Lemma Using Geodesic Homotopiesmentioning

confidence: 99%

Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements

Hardering¹,

Sander²

2014

Found Comput Math

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“…This is not true for the Sobolev space W k,p (Ω, M). The most common definition (see, e.g., [23,20,22,46]) for Sobolev spaces with Riemannian manifold codomains uses the Nash embedding theorem [33]. is independent of ι (see, e.g., [23]).…”

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confidence: 99%

$$L^{2}$$L2-discretization error bounds for maps into Riemannian manifolds

Hardering

2018

Numer. Math.

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We study the approximation of functions that map a Euclidean domain Ω ⊂ R d into an n-dimensional Riemannian manifold (M,g) minimizing an elliptic, semilinear energy in a function set H ⊂ W 1,2 (Ω,M). The approximation is given by a restriction of the energy minimization problem to a family of conforming finite-dimensional approximations S h ⊂ H. We provide a set of conditions on S h such that we can prove a priori W 1,2 -and L 2approximation error estimates comparable to standard Euclidean finite elements. This is done in an intrinsic framework, independently of embeddings of the manifold or the choice of coordinates. A special construction of approximations -geodesic finite elements-is shown to fulfill the conditions, and in the process extended to maps into the tangential bundle. Key words and phrases. Geometric finite elements, L 2 -error bounds, vector field interpolation Energy minimizing maps into and between Riemannian manifolds arise in many contexts, both theoretical and applied. Existence results for harmonic maps have consequences for curvature and topology [23]. Isoperimetric regions (minimizing the area functional), for example, with large volume center in manifolds asymptotic to Schwarzschild have been explored in the context of general relativity and the ADM mass [16]. In general, the influence of an ambient geometry has been of growing interest in the context of geometric flows like mean curvature flow, Ricci flow and Willmore flow. More applied examples are the modelling of oriented materials in Cosserat theory [34], liquid crystals [1], and micromagnetics [29]. Manifold valued harmonic map heat-flow has also been introduced as a regularization in image processing [47].The research interest extends beyond energy minimization in ambient Riemannian manifolds. In ambient spacetimes, spacelike hypersurfaces with vanishing mean curvature are maximizers of the area functional and play a role in general relativity as initial data for solving the Einstein equations. When considering limits of smooth manifolds or problems in optimal transport, the manifold structure of the ambient space needs to be replaced by the general framework of metric spaces. While these prospects are certainly interesting, we will limit our attention to smooth Riemannian target manifolds in the hope that later adaptations can provide a more general theory.

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“…Here π k stands for the homotopy group and [p] is the integral part of p. This result is relatively easy to prove (see also [14,11]). Bethuel [2] proved that in the local case (mappings from a ball) this condition is also sufficient.…”

Section: Introductionmentioning

confidence: 85%

Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces

Hajłasz¹,

Schikorra²

2014

Ann. Acad. Sci. Fenn. Math.

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Abstract. We construct a smooth compact n-dimensional manifold Y with one point singularity such that all its Lipschitz homotopy groups are trivial, but Lipschitz mappings Lip (S n , Y ) are not dense in the Sobolev space W 1,n (S n , Y ). On the other hand we show that if a metric space Y is Lipschitz (n − 1)-connected, then Lipschitz mappings Lip (X, Y ) are dense in N 1,p (X, Y ) whenever the Nagata dimension of X is bounded by n and the space X supports the p-Poincaré inequality. Lipschitz (n − 1)-connectedness is a stronger condition than vanishing of the first n − 1 Lipschitz homotopy groups as it assumes quantitative estimates of Lipschitz constants.

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Sobolev Mappings between Manifolds and Metric Spaces

Cited by 36 publications

References 78 publications

Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements

Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements

$$L^{2}$$L2-discretization error bounds for maps into Riemannian manifolds

Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces

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