A monotonicity formula for stationary biharmonic maps

Angelsberg, Gilles

doi:10.1007/s00209-005-0848-z

Cited by 31 publications

(35 citation statements)

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“…The bound (29) is a consequence of the monotonicity inequality for stationary biharmonic maps. The latter was formally derived by Chang et al [3], Proposition 3.2; for stationary biharmonic maps of class H 2 a rigorous derivation of this key result was later given by Angelsberg [2]. In Appendix B we show how the bound (29) may be derived from the monotonicity inequality directly, without further use of the biharmonic map system.…”

Section: Lemma 33mentioning

confidence: 94%

Partial regularity for biharmonic maps, revisited

Struwe

2008

Calc. Var.

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Section: Lemma 33mentioning

confidence: 94%

Partial regularity for biharmonic maps, revisited

Struwe

2008

Calc. Var.

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“…It was proved first in [4] for sufficiently regular maps and for the general case in [1]. It was proved first in [4] for sufficiently regular maps and for the general case in [1].…”

Section: Facts From the Regularity Theory For Biharmonic Mapsmentioning

confidence: 95%

Dimension reduction for the singular set of biharmonic maps

Scheven¹

2008

Advances in Calculus of Variations

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We consider extrinsically biharmonic maps from an open domain R m into an arbitrary compact submanifold N R L . We show that the singular set of a minimizing biharmonic map has Hausdorff dimension at most m 5. Furthermore, we give conditions on the target manifold under which the dimension can be reduced further and conditions under which similar results hold for maps that are only stationary biharmonic or stable stationary biharmonic. The proof is based on the analysis of defect measures by tools of geometric measure theory. A byproduct of the proof is the theorem that weak limits of stationary biharmonic maps are weakly biharmonic.for all variations of the form u t D N .u C tW / with W 2 C 1 cpt .; R L /, where N W U ı ! N denotes the nearest-point-retraction from a neighborhood U ı R L of N onto N . Smooth biharmonic maps are characterized by 2 u ? N on , cf. (2.1).A weakly biharmonic map u 2 W 2;2 .; N / is called stationary biharmonic if it satisfies (1.1) additionally for variations of the domain, that is u t .x/ D u.x C t .x// with 2 C 1 cpt .; R m /.

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“…However, they derived this monotonicity formula only for sufficiently regular maps. G. Angelsberg [1] gave a rigorous proof of this monotonicity formula for stationary biharmonic maps u ∈ W 2,2 (B r , N ). A monotonicity formula for intrinsically stationary biharmonic maps was derived by R. Moser [11].…”

Section: Introductionmentioning

confidence: 98%

A boundary monotonicity inequality for variationally biharmonic maps and applications to regularity theory

Altuntas

2018

Ann Glob Anal Geom

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We derive a boundary monotonicity formula for a class of biharmonic maps with Dirichlet boundary conditions. A monotonicity formula is crucial in the theory of partial regularity in super-critical dimensions. As a consequence of such a boundary monotonicity formula, one is able to show partial regularity for variationally biharmonic maps and full boundary regularity for minimizing biharmonic maps. *

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A monotonicity formula for stationary biharmonic maps

Abstract: Abstract. We give a rigorous proof of the monotonicity formula of S.-Y.A. Chang, L. Wang and P. Yang [3] for (extrinsically) stationary biharmonic maps of class W 2,2 .

Cited by 31 publications

References 11 publications

Partial regularity for biharmonic maps, revisited

Partial regularity for biharmonic maps, revisited

Dimension reduction for the singular set of biharmonic maps

A boundary monotonicity inequality for variationally biharmonic maps and applications to regularity theory

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