Preliminary I will expose a technique developed with T. Rivi\`{e}re to prove energy identities (weak compactness) for sequences of solutions of any conformally invariant problem of second order in dimension 2, see [1]. Then after introducing some basic facts about Willmore surfaces(a fourth order problem), I will prove a general energy identity for them, see [2]. I will notably focus on the case when the conformal class degenerates and I will also explain the crucial role played by some residues in the collar region and some estimates on Green functions, see [3].

[1] P. Laurain, T. Rivière, Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications. Anal. PDE, 7(1):1–41, 2014.

[2] P. Laurain, T. Rivière, Energy quantization of Willmore surfaces at the boundary of the moduli space. Duke Math. J. 167 (2018), no. 11, 2073–2124.

[3] P. Laurain, T. Rivière, Optimal estimate for the gradient of Green functions on degenerating surfaces and applications, Comm. Anal. Geom. Vol. 26, No 4, 2018.

# Analysis of some Conformally Invariant Problems

Paul Laurain

Univeristé Paris Diderot

October 30, 2018