Mapping problems, fundamental groups and defect measures (Q1284787)

The paper deals with the limiting behaviour of minimizing sequences for the \(p\)-harmonic problem \[ \inf\biggl\{\int_M | \nabla u| ^p dx: u:M\to N\text{continuous, \(u=g\) on \(\partial M\)} \biggr\} \] where \(M, N\) are smooth Riemannian manifolds, with \(\partial N=0\). Two are the main results of the paper: the first result is that any minimizing sequence \((u_k)\) weakly converges (up to subsequences) to a \(p\)-harmonic map \(u:M\to N\), smooth away from a \((n-p)\)-rectifiable closed subset \(\Sigma\) of \(M\) with finite \({\mathcal H}^{n-p}\) measure (where \(n=\dim(M)\)). Moreover, the defect energy \(\lim_k| \nabla u_k| ^2dx\) is representable as \(| \nabla u| ^2dx+\nu\) with \(\nu\), the so-called defect energy, supported on \(\Sigma\). The second result is that if \(p\in (1,n)\) is not an integer then the defect measure vanishes, hence \((u_j)\) strongly converge to \(u\). The proofs are based on several blow-up techniques introduced by the author in previous papers, together with Marstrand density theorem. A new technical ingredient is also the theory of generalized varifolds, first introduced by \textit{L. Ambrosio} and \textit{H. M. Soner} [Prog. Nonlinear Differ. Equ. Appl. 25, 123-134 (1996; Zbl 0878.35053)] in connection with mean curvature flows of surfaces of codimension higher than one.