Elsevier

Nonlinear Analysis

Volume 200, November 2020, 111971
Nonlinear Analysis

Blow-up analysis and boundary regularity for variationally biharmonic maps

https://doi.org/10.1016/j.na.2020.111971Get rights and content

Abstract

We consider critical points u:ΩN of the bi-energy Ω|Δu|2dx,where ΩRm is a bounded smooth domain of dimension m5 and NRL a compact submanifold without boundary. More precisely, we consider variationally biharmonic maps uW2,2(Ω,N), which are defined as critical points of the bi-energy that satisfy a certain stationarity condition up to the boundary. For weakly convergent sequences of variationally biharmonic maps, we demonstrate that the only obstruction that can prevent the strong compactness up to the boundary is the presence of certain non-constant biharmonic 4-spheres or 4-halfspheres in the target manifold. As an application, we deduce full boundary regularity of variationally biharmonic maps provided such spheres do not exist.

Section snippets

Introduction and statement of the results

Biharmonic maps are a higher order variant of harmonic maps uC(Ω,N) into a Riemannian manifold NRL, which are defined as critical points of the Dirichlet energy E1(u)Ω|Du|2dx.Analogously, we call a map uC(Ω,N) biharmonic if it is a critical point of the bi-energy E2(u)Ω|Δu|2dx.More generally, maps uW2,2(Ω,N) that satisfy the Euler–Lagrange equation of E2 in the weak sense are called weakly biharmonic, and the class of weakly harmonic maps is defined accordingly. The analytical and

Notation

We write Br(a)Rm for the open ball with radius r>0 and center aRm and Sr(a)Br(a) for the corresponding sphere. For the upper halfspace, we use the abbreviation R+mRm1×[0,), and write Br+(a)Br(a)R+m for arbitrary centers aR+m. Moreover, in the case of a center a=(a,0)R+m we use the abbreviations Sr+(a)Sr(a)R+m for the curved part and Tr(a)Br(a)×{0} for the flat part of the boundary of Br+(a). 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 uiW2,2(B4+,N) with respect to Dirichlet values giC4,α(B4+,N) on T4. We assume that the sequence is bounded in the sense that supiN(D2uiL2(B4+)+DuiL2(B4+))<andsupiNgiC4,α(B4+)<.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 W2,2. 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 v:S+m1N that are biharmonic with respect to a certain Paneitz-bi-energy, cf. [21, Lemma 5.1]. We note that the case m=5, which corresponds to Paneitz-biharmonic 4-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 uW2,2(B2+,N) in a boundary point aT1. For r(0,1) we define the rescaled maps ua,rW2,2(B1r+,N)   by   ua,r(x)u(a+rx).A map vWloc2,2(R+m,N) is called a tangent map of u at the point aT1 if there is a sequence ri0

Acknowledgments

This work has been supported by the DFG, Germany -project SCHE 1949/1-1 “Randregularität biharmonischer Abbildungen zwischen Riemann’schen Mannigfaltigkeiten”.

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