Generalized elastica problems under area constraint

It was recently proved in [D. Bucur and A. Henrot, J. Eur. Math. Soc. (JEMS) 19, No. 11, 3355–3376] and [V. Ferone, B. Kawohl & C. Nitsch, Math. Ann., 365, No. 3–4, 987–1015] that the elastic energy E(γ)=1/2int_γ κ^2 ds of a closed curve γ with curvature κ has a minimizer among all plane, simple, regular and closed curves of given enclosed area A(γ), and that the minimum is attained only for circles. In particular, the proof used in [V. Ferone, B. Kawohl & C. Nitsch, loc. cit.] is of a geometric nature, and here, we show under which hypothesis it can be extended to other functionals involving the curvature. As an example, we show that the optimal shape remains a circle for the p-elastic energy int_γ |κ|^p ds, whenever p>1.