Estimates for the energy density of critical points of a class of conformally invariant variational problems (Q2861481)

The article establishes an estimate for the energy density of critical points of a large class of conformally invariant variational problems with small energy on the two-dimensional unit disc. In particular, it is shown that the energy density of the critical points lies in the local Hardy space \(h_1(B_1)\). The estimate can be applied to the harmonic map equation, the prescribed mean curvature equation in \(\mathbb R^3\) and, more generally, to every Euler-Lagrange equation of any conformally elliptic Lagrangian being quadratic in the gradient. NEWLINENEWLINENEWLINEAs a corollary a new proof of the energy convexity and uniqueness result for weakly harmonic maps with small energy on the unit disc is obtained. NEWLINENEWLINENEWLINETo prove the statement, the authors exploit the special form of the Euler-Lagrange equation, which has a Jacobian determinant structure. In addition, a refined version of the techniques from \textit{T. Rivière} [Invent. Math. 168, No. 1, 1--22 (2007; Zbl 1128.58010)] is used.