Conservation laws for conformally invariant variational problems

Abstract

Theorem II.1 [Mor1] Let u be a map in the Sobolev space C 0 ∩ W 1,2 (D 2 , R m ) and let ε > 0, then there exists an homeomorphism Ψ of the disc such that Ψ ∈ W 1,2 (D 2 , D 2 ),

Estimates for the energy density of critical points of a class of conformally invariant variational problems

“…Following the strategy of Uhlenbeck in [Uh82], Rivière [Ri07] used the algebraic feature of Ω, namely Ω being antisymmetric, to construct ξ ∈ W 1,2 0 (B 1 , so(n)) and a gauge transformation matrix P ∈ W 1,2 ∩L ∞ (B 1 , SO(n)) (which pointwise almost everywhere is an orthogonal matrix in R n×n ) satisfying some good properties.…”

“…Later, Qing [Qi93] showed continuity up to the boundary in the case of continuous boundary data based on Hélein's technique. More recently, Rivière [Ri07] Here and throughout the paper, the Einstein summation convention is used. In particular, this special form of the nonlinearity enabled Rivière to obtain a conservation law for this system of PDE's (see (3.7) below), which is accomplished via a technique that we call Rivière's gauge decomposition, see Section 3.…”

n/p-harmonic maps: Regularity for the sphere case

“…That is, the highest order terms scale exactly as the lowerorder terms, thus inhibiting the application of a general regularity theory based only on the general growth of the right-hand side -one has to consider the finer behavior of the equation: These exhibit an antisymmetric structure, which is closely related to the appearance of Hardy spaces and compensated compactness -and induces regularity of critical points. In two dimensions, these facts were observed in Rivière's celebrated [Riv07] for all conformally invariant variational functionals (of which the Dirichlet energy is a prototype). We refer the interested reader to the introductions of [DLR11b], [DLR11a] for more on this.…”

Boundary regularity via Uhlenbeck–Rivière decomposition

“…The interior regularity was proved by T. Rivière in [Riv07] (for e ≡ 0) and our proof is based on Rivière's decomposition result combined with the Dirichlet growth approach by Rivière and Struwe in [RS08] as well as some additional arguments due to P. Strzelecki [Str03]. Remark 1.3 Let us emphasize that one can prove Theorem 1.1 also by reflection across ∂D 2 , whenever there is some ψ ∈ W 2,p (D 2 , R m ), p > 1, such that u = ψ on ∂D 2 .…”