Abstract. We give a rigorous proof of the monotonicity formula of S.-Y.A. Chang, L. Wang and P. Yang [3] for (extrinsically) stationary biharmonic maps of class W 2,2 .

We prove a regularity result for critical points of the polyharmonic energywith k ∈ N and p > 1. Our proof is based on a Gagliardo-Nirenberg-type estimate and avoids the moving frame technique. In view of the monotonicity formulae for stationary harmonic and biharmonic maps, we infer partial regularity in theses cases.

We seek critical points of the Hessian energy functional E (u) = | u| 2 dx, where = R 4 or is the unit disk B in R 4 and u : → S 4 . We show that E R 4 has a critical point which is not homotopic to the constant map. Moreover, we prove that, for certain prescribed boundary data on ∂B, E B achieves its infimum in at least two distinct homotopy classes of maps from B into S 4 .

Law invariant convex risk measures, Robust representation, Variational methods, 91G99, D81, G22,