Boundary regularity via Uhlenbeck–Rivière decomposition

Müller, Frank; Schikorra, Armin

doi:10.1524/anly.2009.1025

Cited by 22 publications

(15 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular when f ≡ 0, (2) describes harmonic maps and prescribed mean curvature equations from Riemannian surfaces into closed, C 2 Riemannian manifolds N → R m isometrically embedded in some Euclidean space. This PDE has subsequently been studied from a regularity and compactness perspective, see for instance [RL13], [LZ09], [MS09], [Sch10], [ShTo13]. In [ShTo13], it is shown that general solutions to (2) (when n = 2) are in W …”

mentioning

confidence: 99%

Higher integrability for solutions to a system of critical elliptic PDE

Sharp

2014

Methods and Applications of Analysis

View full text Add to dashboard Cite

Abstract. We give new estimates for a critical elliptic system introduced by Rivière-Struwe which generalises PDE solved by (almost) harmonic maps from a Euclidean ball B 1 ⊂ R n into closed Riemannian manifolds N → R m . Solutions u : B 1 → R m take the formwhere Ω maps into antisymmetric m × m matrices with entries in R n . Here Ω and ∇u belong to a Morrey space which makes the PDE critical from a regularity perspective. We use the Coulomb frame method employed by Rivière-Struwe along with the Hölder regularity already acquired therein, coupled with an extension of a Riesz potential estimate, in order to improve the known regularity and estimates for solutions u. These methods apply when n = 2 thereby re-proving the full regularity in this case using Coulomb gauge methods. Moreover they lead to a self contained proof of the local regularity of stationary harmonic maps in high dimension.

show abstract

mentioning

confidence: 99%

Higher integrability for solutions to a system of critical elliptic PDE

Sharp

2014

Methods and Applications of Analysis

View full text Add to dashboard Cite

show abstract

“…To proceed, we recall that the regularity up to the boundary for weak solutions satisfying (2.21) with continuous boundary trace was established by Müller-Schikorra [22]. More precisely, they proved that Theorem C. Let D ⊂ R 2 be a simply connected domain with C 2 boundary ∂D.…”

Section: Qun Chen Jürgen Jost Guofang Wang Miaomiao Zhumentioning

confidence: 95%

“…Our methods will also utilize the general strategy of Rivière [24] who had achieved an important generalization of the earlier results of Wente [27] and Hélein [14,15]. Rivière's approach has been adapted to Dirichlet boundary regularity by Müller-Schikorra [22], and this work will also be useful for our purposes.…”

Section: Qun Chen Jürgen Jost Guofang Wang Miaomiao Zhumentioning

confidence: 99%

The boundary value problem for Dirac-harmonic maps

Chen¹,

Jost²,

Wang³

et al. 2013

J. Eur. Math. Soc.

View full text Add to dashboard Cite

Abstract. Dirac-harmonic maps are a mathematical version (with commuting variables only) of the solutions of the field equations of the non-linear supersymmetric sigma model of quantum field theory. We explain this structure, including the appropriate boundary conditions, in a geometric framework. The main results of our paper are concerned with the analytic regularity theory of such Dirac-harmonic maps. We study Dirac-harmonic maps from a Riemannian surface to an arbitrary compact Riemannian manifold. We show that a weakly Diracharmonic map is smooth in the interior of the domain. We also prove regularity results for Dirac-harmonic maps at the boundary when they solve an appropriate boundary value problem which is the mathematical interpretation of the D-branes of superstring theory.

show abstract

“…It is natural to investigate the regularity up to the boundary for these kind of geometric equations. In [37] Müller and the author considered the Dirichlet problem and showed Theorem 1.2 (Müller-S. [37]). Let D be a smoothly bounded domain.…”

Section: Introductionmentioning

confidence: 99%

“…Continuity up to the boundary follows now from [37]. Hölder continuity follows from a reflection argument: Since u is Hölder continuous, so is V. Thus, for any x 0 ∈ (−1, 1)×{0},…”

mentioning

confidence: 99%

Boundary Equations and Regularity Theory for Geometric Variational Systems with Neumann Data

Schikorra

2018

Arch Rational Mech Anal

Self Cite

View full text Add to dashboard Cite

We study boundary regularity of maps from two-dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which, at the boundary, enter perpendicularly into a support manifold. For example, harmonic maps, or H-surfaces, with a partially free boundary condition.In the interior it is known, by the celebrated work of Rivière, that these maps satisfy a system with an antisymmetric potential, from which one can derive regularity of the solution. We show that these maps satisfy along the boundary a system with a nonlocal antisymmetric boundary potential which contains information from the interior potential and the geometric Neumann boundary condition. We then proceed to show boundary regularity for solutions to such systems.

show abstract

Boundary regularity via Uhlenbeck–Rivière decomposition

Cited by 22 publications

References 17 publications

Higher integrability for solutions to a system of critical elliptic PDE

Higher integrability for solutions to a system of critical elliptic PDE

The boundary value problem for Dirac-harmonic maps

Boundary Equations and Regularity Theory for Geometric Variational Systems with Neumann Data

Contact Info

Product

Resources

About