Abstract
In this paper we consider Lipschitz graphs of functions which are stationary points of strictly polyconvex energies. Such graphs can be thought as integral currents, resp. varifolds, which are stationary for some elliptic integrands. The regularity theory for the latter is a widely open problem, in particular no counterpart of the classical Allard’s theorem is known. We address the issue from the point of view of differential inclusions and we show that the relevant ones do not contain the class of laminates which are used in [23] and [26] to construct nonregular solutions. Our result is thus an indication that an Allard’s type result might be valid for general elliptic integrands. We conclude the paper by listing a series of open questions concerning the regularity of stationary points for elliptic integrands.
Highlights
Let Ω ⊂ Rm be open and f ∈ C1(Rn×m, R) be a polyconvex function, i.e. such that there is a convex g ∈ C1 such that f (X) =g(Φ(X)), where Φ(X) denotes the vector of subdeterminants of X of all orders
The main result of the present paper shows that such strategy fails in the case of stationary points
More precisely: (a) We show that u solves (1.2), (1.3) if and only if there exists an L∞ matrix field A that solves a certain system of linear, constant coefficients, PDEs and takes almost everywhere values in a fixed set of matrices, which we denote by Kf and call inclusion set, cf
Summary
A first interesting question is whether one could extend the examples of Müller and Šveràk and Székelyhidi to provide counterexamples Both in [23] and [26], the starting point of the construction of irregular solutions is rewriting the condition (1.2) as a differential inclusion, and finding a so-called TN -configuration In particular [26, Theorem 1] shows the existence of a smooth strongly polyconvex integrand f ∈ C∞(R2×2) for which the corresponding “classical” differential inclusion contains a T5 configuration (cf Definition 2.5). A map u ∈ Lip(Ω, Rn) is a stationary point of the energy (1.1) if and only there are matrix fields Y ∈ L∞(Ω, Rn×m) and Z ∈ L∞(Ω, Rm×m) such that W = (Du, Y, Z) solves the div-curl differential inclusion (2.2) and (2.3)
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