MiS Preprint Repository

Let $\Omega \subset \mathbb{R}^2$ be a bounded Lipschitz domain and let $F:\Omega \times \mathbb{R}^{2 \times 2}_+ \to \mathbb{R}$ be a Carathèodory integrand such that $F(x,\cdot)$ is polyconvex for ${L}^2$- a.e. $x \in \Omega$. Moreover assume that F is bounded from below and satisfies the condition $F(x, \xi) \to \infty$ as $\det \xi \to 0^+$ for $L^2$- a.e. $x \in \Omega$. In this article we study the effect of domain topology on the existence and multiplicity of strong local minimizers of the functional $$\Gamma [u] := \int_\Omega F(x,\nabla u (x))dx$$ where the map u lies in the Sobolev space $W^{1,p}_{id} (\Omega , \mathbb{R}^2)$ with $p \geq 2$ and satisfies the pointwise condition $\det \nabla u(x)>0$ for $L^2$-a.e. $x \in \Omega$. We settle the question by establishing that $\mathbb{F} [\cdot]$ admits a set of strong local minimizers on $W^{1,p}_{id} (\Omega , \mathbb{R}^2)$ that can be indexed by the group $\mathbb{P}_n \bigoplus \mathbb{Z}^n$, the direct sum of Artin's pure braid group on n strings and n copies of the infinite cyclic group. The dependence on the domain topology is through the number of holes n in $\Omega$ and the different mechanisms that give rise to such local minimizers are fully exploited by this particular representation.