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

Abstract

Let $F\in W_{loc}^{1,n}(Ω;\Bbb R^n)$ be a mapping with non-negative Jacobian $J_F(x)=\text{det} DF(x)\ge 0$ a.e. in a domain $Ω\in \Bbb R^n$. The dilatation of the mapping $F$ is defined, almost everywhere in $Ω$, 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 Rešhetnyak'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 Šverak [IS] have shown that $K(x)\in L^1_{loc}$ is sufficient to guarantee the conclusion of Rešhetnyak'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 $Ω$. 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.