Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals

Abstract

Abstract. We give a new proof of the small excess regularity theorems for integer multiplicity recti able currents of arbitrary dimension and codimension minimizing an elliptic parametric variational integral. This proof does not use indirect blow-up arguments, it covers interior and boundary regularity, it applies to almost minimizing currents, and it gives an explicit and often optimal modulus of continuity for the derivative, i.e. for the tangent plane eld of the almost minimizing currents. IntroductionThe regularity theory for integer multiplicity recti able currents minimizing a parametric elliptic variational integral was initiated in the pioneering work of F.J. Almgren Alm1 , where the regularity is proved at interior support points with small cylindrical excess. Almgren's results are presented in F. Federer's monograph Fe, Chap. 5 . E. Bombieri has given a somewhat simpler proof which also applies to approximately minimizing currents. This is important in order to be abble to include problems with side conditions, such as volume constraints or obstacle conditions, in the regularity theory see Section 1 . In Alm2 Almgren also treated the more general situation of almost minimizing currents and sets. A new and elegant proof of Almgren's original regularity theorem was presented by R . S c hoen and L. Simon SS . This proof avoids the indirect blow-up arguments of the previous authors and it uses a direct P.D.E. argument instead. The interior regularity theory for almost minimizing currents has been thus quite well developed for some years.With regard to boundary regularity the situation is di erent. One only has the work of R. Hardt Ha1 which showed how to extend the reasoning from Fe, Chap. 5.3 to the boundary situation. Apart from Ha1 , we are not aware of any published treatment of boundary regularity for almost minimizing currents and general elliptic integrands. This problem is addressed, however, in the unpublished thesis Li of F.H. Lin.Our present work comes from an attempt to understand the question of boundary regularity better and to give a simpler proof of Hardt's boundary regularity result, and if possible to extend it to the case of almost minimizing currents. We rst tried to carry over the simple method of SS to the boundary situation. This failed, however, for reasons explained below. Instead we h a ve combined arguments from Bo with the idea of harmonic approximation as presented in Si1, 21.1 and we h a ve added some new estimates related to the integrands one obtains from a given smooth integrand by transforming it with di eomorphisms that atten the boundary. As a result, we h a ve obtained small excess regularity theorems for almost minimizing currents which give an optimal modulus of continuity for the tangent plane eld in many situations. It seems that even in the interior situation suchWe are grateful to the Volkswagen-Stiftung and to the Oberwolfach Mathematical Research Institute whose support through the RiP collaborative research program made the commencement of this work possible. ...

A Simple Partial Regularity Proof for Minimizers of Variational Integrals

“…Since we will use A-harmonic functions as comparison maps later, we will now state an approximation result, which includes some a priori estimates for A-harmonic functions. It has been stated in [39,Lemma 6.8] (compare [7]) in the present form and is a variant of the A-harmonic approximation lemma of Duzaar & Steffen [9]. The reader should note that the definition of V in [39] is different, but the lemma is easily seen to hold also with the present definition (2.1).…”

“…where c depends only on p. In the remainder of this section we adapt essentially the arguments of [5,6,8,9]. We will use Lemma 4.3 and Lemma 5.1 to derive decay estimates for the excess of the minimizer.…”

“…In fact, we apply the A-harmonic approximation method of Duzaar & Steffen [9] and bypass the use of Gehring's lemma in the proof of a Caccioppoli type estimate by a subtle, but elementary splitting of the relevant terms. All in all we will provide an elementary, self-contained, and comparably short proof of Theorem 1.1.…”

The quantitative isoperimetric inequality and related topics

“…Beside the technical complications due to the fact that one is now dealing with higher codimension, the main ingredients come from the theory of currents that 'almost' minimize certain elliptic variational integrals. The regularity results needed in the proof were obtained in [20,54].…”

Regularity of Free Boundaries in Anisotropic Capillarity Problems and the Validity of Young’s Law

“…Note that if cl(A) ⊂ H (as it happens, for example, in the limit case H = R n ), then Definition 1.8 reduces to a local almost-minimality notion analogous to the ones considered in [Alm76,Bom82,Tam84,DS02] and [Mag12,Section 21].…”