MiS Preprint Repository

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