Blow-up analysis and boundary regularity for variationally biharmonic maps
Section snippets
Introduction and statement of the results
Biharmonic maps are a higher order variant of harmonic maps into a Riemannian manifold , which are defined as critical points of the Dirichlet energy Analogously, we call a map biharmonic if it is a critical point of the bi-energy More generally, maps that satisfy the Euler–Lagrange equation of in the weak sense are called weakly biharmonic, and the class of weakly harmonic maps is defined accordingly. The analytical and
Notation
We write for the open ball with radius and center and for the corresponding sphere. For the upper halfspace, we use the abbreviation , and write for arbitrary centers . Moreover, in the case of a center we use the abbreviations for the curved part and for the flat part of the boundary of . If the center is clear from the context, we will often omit it in the notation and
The defect measure
Our goal is to prove compactness for a sequence of variationally biharmonic maps with respect to Dirichlet values on . We assume that the sequence is bounded in the sense that More precisely, we consider the slightly more general case of maps with (1), (6) instead of variationally biharmonic maps, since the properties (1), (6) are clearly preserved under strong convergence in . In view of (27),
Liouville type theorems for biharmonic maps on a half space
The next theorem excludes the existence of certain non-constant biharmonic maps which might occur as tangent maps in singular boundary points. We remark that since these maps are homogeneous of degree zero, they can also be interpreted as maps that are biharmonic with respect to a certain Paneitz-bi-energy, cf. [21, Lemma 5.1]. We note that the case , which corresponds to Paneitz-biharmonic -halfspheres, cannot be treated with the same methods and will be postponed to Lemma 4.3.
Theorem 4.1 Let
Properties of tangent maps
The strategy for the proof of the full boundary regularity is to apply the dimension reduction argument by Federer in order to prove that the dimension of the singular set is zero. To this end, we first provide some properties of tangent maps of a variationally biharmonic map in a boundary point . For we define the rescaled maps A map is called a tangent map of at the point if there is a sequence
Acknowledgments
This work has been supported by the DFG, Germany -project SCHE 1949/1-1 “Randregularität biharmonischer Abbildungen zwischen Riemann’schen Mannigfaltigkeiten”.
References (31)
An optimal partial regularity result for minimizers of an intrinsically defined second-order functional
Ann. Inst. H. Poincaré Anal. Non Linéaire
(2009)A boundary monotonicity inequality for variationally biharmonic maps and applications to regularity theory
Ann. Global Anal. Geom.
(2018)A monotonicity formula for stationary biharmonic maps
Math. Z.
(2006)On the singular set of stationary harmonic maps
Manuscripta Math.
(1993)- et al.
A regularity theory of biharmonic maps
Comm. Pure Appl. Math.
(1999) - et al.
On a fourth order curvature invariant
- et al.
Two Reports on Harmonic Maps
(1995) - et al.
Boundary partial regularity for a class of biharmonic maps
Calc. Var. Partial Differential Equations
(2012) - et al.
Boundary regularity for polyharmonic maps in the critical dimension
Adv. Calc. Var.
(2009) Applications harmoniques de surfaces riemanniennes
J. Differential Geom.
(1978)