Discreteness and openness for mappings of finite distortion in the critical case $p=n-1$

Abstract

Let $F\in W_{loc}^{1,n}(\Omega;\Bbb R^n)$ be a mapping with non-negative Jacobian $J_F(x)=\text{det} DF(x)\ge 0$ a.e. in a domain $\Omega\in \Bbb R^n$. The dilatation of the mapping $F$ is defined, almost everywhere in $\Omega$, by the formula $$K(x)={{|DF(x)|^n}\over {J_F(x)}}.$$ If $K(x)$ is bounded a.e., the mapping is said to be quasiregular. Quasiregular mappings are a generalization to higher dimensions of holomorphic mappings. The theory of higher dimensional quasiregular mappings began with Re\v{s}hetnyak's theorem, stating that non constant quasiregular mappings are continuous, discrete and open. In some problems appearing in the theory of non-linear elasticity, the boundedness condition on $K(x)$ is too restrictive. Tipically we only know that $F$ has finite dilatation, that is, $K(x)$ is finite a.e. and $K(x)^p$ is integrable for some value $p$. In two dimensions, Iwaniec and \v{S}verak [IS] have shown that $K(x)\in L^1_{loc}$ is sufficient to guarantee the conclusion of Re\v{s}hetnyak's theorem. For $n\ge 3$, Heinonen and Koskela [HK], showed that if the mapping is quasi-light and $K(x)\in L^p_{loc}$ for $p>n-1$, then the mapping $F(x)$ is continuous, discrete and open. Manfredi and Villamor [MV] proved a similar result without assuming that the mapping $f(x)$ was quasi-light. The result is known to be false, see [Ball], when $p<n-1$. In this paper we attempt to improve in those results. In particular, we will deal with the case $p=n-1$ for $n\ge 3$, and will assume that our mapping $F(x)$ is quasi-light, that is, the inverse image of any point is compact in $\Omega$. Our approach will be different from the ones used in [MV] and [HK]. It is more geometrical in nature and uses the method of extremal length.