Journal ArticleDOI
Partial regularity for stationary harmonic maps into spheres
TLDR
In this paper, it was shown that any weakly harmonic mapping from a two-dimensional surface into a sphere is smooth, except possibly for a closed singular set of (n 2 ) dimensional Hausdorff measure zero.Abstract:
In an interesting recent paper [12], F. HI~LEIN has shown that any weakly harmonic mapping from a two-dimensional surface into a sphere is smooth. I present here a kind of generalization to higher dimensions, asserting in effect that a stationary harmonic mapping from an open subset of Nn(n __> 3) into a sphere is smooth, except possibly for a closed singular set of (n 2 ) dimensional Hausdorff measure zero. My proof crucially depends upon several of H~L~IN's observations (as streamlined by P.-L. LIoNs). To state the result precisely let us suppose that m, n => 2, U is a smooth open subset of Nn, and S m-1 denotes the unit sphere in Nm. A function u in the Sobolev space HI (U;Rm) , u = (u 1 . . . . ,urn), belongs to H I ( U ; S ml ) provided t u[ = 1 a.e. in Uread more
Citations
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Book
An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs
TL;DR: In this article, the authors focus on the regularity theory for elliptic systems and illustrate some of the basic ideas and techniques introduced in this context, confining themselves to important but simple situations and refraining from completeness.
Journal ArticleDOI
The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations
TL;DR: In this article, the critical Sobolev inequalities in the Besov spaces with the logarithmic form such as Brezis-Gallouet-Wainger and Beale-Kato-Majda were studied.
Journal ArticleDOI
Conservation laws for conformally invariant variational problems
TL;DR: In this paper, a 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations,..., etc.) is presented in divergence form.
Journal ArticleDOI
On the singular set of stationary harmonic maps.
TL;DR: In this article, it was shown that a weakly harmonic map from a compact riemannian manifold to a stationary harmonic map is smooth in the special case that the manifold is a sphere.
References
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BookDOI
Multiple integrals in the calculus of variations
TL;DR: In this paper, a variational method in the theory of harmonic integrals has been proposed to solve the -Neumann problem on strongly pseudo-convex manifolds and parametric Integrals two-dimensional problems.
Journal ArticleDOI
H p spaces of several variables
BookDOI
Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105), Volume 105
TL;DR: In this article, the authors present a book, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems (AM-105), which is an extension of their previous work.
Journal ArticleDOI
A regularity theory for harmonic maps
Richard Schoen,Karen Uhlenbeck +1 more
TL;DR: In this paper, it was shown that a bounded, energy minimizing map u: M -N is regular (in the interior) except for a closed set S of Hausdorff dimension at most n − 3.0.