Minima of Invariant Functions: The Inverse Problem
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Minima of Invariant Functions: The Inverse Problem Jürgen Scheurle · Sebastian Walcher
Received: 6 July 2014 / Accepted: 3 September 2014 © Springer Science+Business Media Dordrecht 2014
Abstract We determine locally minimizing functions that are invariant with respect to the action of a finite linear group. This resolves a problem which is inverse to one discussed in a seminal paper by Abud and Sartori, and occurs naturally in various physical applications, such as elasticity theory and phase transitions. A general existence result reduces the local problem to elementary computations. Some results are extended to the compact case, and some examples and applications are given. Keywords Symmetry breaking · Orbit space · Elasticity · Shape-memory alloys Mathematics Subject Classification 13A50 · 58K70 · 74B05 · 74D05 1 Introduction and Overview Minima of functions that are invariant under a (compact) group action have been in the focus of theoretical as well as applied physics for several decades. In particular, symmetry breaking at critical points was identified as a crucial mechanism to explain a number of physical phenomena. Several fundamental contributions to the underlying mathematical theory culminated in papers by Michel and Radicati [12], Michel [13], Michel and Zhilinskii [14], and Abud and Sartori [1, 2] (the latter will be our basic reference). Abud and Sartori [2] passed to the orbit space (realized as a semi-algebraic variety via the Hilbert map) and resolved the problem of finding minima on strata (which are submanifolds) via the Lagrange multiplier method. In the present paper, we will discuss the inverse problem; viz. to position a minimum of a G-invariant function at some prescribed point. This inverse problem is relevant, for instance, in the modelling and analysis of strain-energy density functions in
J. Scheurle Zentrum Mathematik, TU München, Boltzmannstr. 3, 85747 Garching, Germany e-mail: [email protected]
B
S. Walcher ( ) Mathematik A, RWTH Aachen, 52056 Aachen, Germany e-mail: [email protected]
J. Scheurle, S. Walcher
elasticity theory (see e.g. Coleman and Noll [5], Smith and Rivlin [17]). For readers who are less familiar with this area of research, we give a brief account of the relevant physical and mathematical background. For materials which are perfectly elastic (hyperelastic), at a given temperature, the strainenergy density is a smooth function over the set of all deformation gradients or rather over all Cauchy-Green strain tensors. The latter are symmetric and positive definite. Critical points of this function correspond to stress-free configurations of the material. At (local) minima, these are stable in the sense that typically lower energies are preferred. If the material possesses symmetry, i.e. the underlying crystal lattice is invariant under certain symmetry transformations, the strain-energy function is invariant under the action of the corresponding symmetry group on the set of all Cauchy-Green strain tensors by conjugacy. This imposes restrictions on
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