𝑄-valued functions revisited

Multi-Value Microstructural Descriptors for Complex Materials: Analysis of Ground States

“…It has been also revisited and exploited in different contexts (see [13,14] for a more detailed bibliography on the subject). …”

“…The right hand side of (4.4) can be estimated by using a change of coordinates and the chain rule for multiple valued functions as proven in [13,Proposition 1.12]. So, we get…”

Rectifiability of the singular set of multiple-valued energy minimizing harmonic maps

“…Fix an integer Q ≥ 1. We will assume that the reader is familiar with the theory of Almgren's Q-valued functions, for which we refer to [DLS11]. In what follows, we briefly recall the main definitions and properties we are going to need in the sequel.…”

“…These differences make the study of such objects both more complicated than their classical counterpart and more interesting. For a recent survey on results of this kind, we direct the reader to [DLS11], where the authors revisit Almgren's original regularity theory of Dir-minimizing Q-valued functions suggesting a more intrinsic approach which has its roots in the new techniques developed in the last two decades to perform analysis on metric spaces.…”

Rectifiability and upper Minkowski bounds for singularities of harmonic $Q$-valued maps

“…[DL16, Section 7.3]). It has been noticed in [DLS11] that modern analysis in metric spaces can be used to give an intrinsic simple definition of both objects. We refer to [DLS11,Alm00] for a more detailed description of the space of Q-valued functions and Dirichlet minimizers, here we simply recall that Dirichlet minimizers are Hölder continuous functions with exponent α = α(m, n, Q).…”