The dirichlet energy of mappings with values into the sphere

Currents in metric spaces

“…Fleming and H. Federer developed in [24] the theory of currents, which leads to existence results for the Plateau problem for oriented surfaces of any dimension and codimension. It is now clear that the interest of this theory, which includes in some sense the theory of Sobolev and BV-functions, goes much beyond the area-minimization problems that were its initial motivation: as an example one can" consider the recent papers [3], [8], [27], [28], [29], [35], [41], [42], to quote just a few examples.…”

Ground states in complex bodies

“…In the case of homogeneous bodies, (5.13) implies the the internal 'power' of the extended Hamilton-Eshelby stress is determined only by the interactions along the line of concentration of energy. Equation (5.13) extends also a theorem in [26] to the case in which macroscopic deformations occur.…”

“…involving maps ν : B 0 → S 2 , with S 2 the unit sphere, and for Dirichlet boundary data (see results in [23,25,26]). Take note that the exponent 2 and the circumstance that the manifold of substructural shapes is S 2 is crucial for the remarks in what follows.…”

Singularities of harmonic maps

“…M. Giaquinta, G. Modica, and Soucêk proved in [44] that sing u is a closed rectifiable set of finite H 1 measure. Their proof involved their important general program [45] of deriving and understanding the relaxed energy F using a completion of the graph of u called a Cartesian current.…”

“…Is the singularity estimate of [44] optimal ? In the restricted context of axially symmetric maps for B 3 to S 2 , Hardt, Lin, and Poon [64] showed that the singular set of an F minimizer is at most an isolated set (in the axis).…”