Mappings of finite distortion: the degree of regularity

Abstract

This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K( x)⩾1 be a measurable function defined on a domain Ω⊂ R n, n⩾2 , and such that exp(βK(x))∈L loc 1(Ω) , β>0. We show that there exist two universal constants c 1( n), c 2( n) with the following property: Let f be a mapping in W loc 1,1(Ω, R n) with | Df( x)| n ⩽ K( x) J( x, f) for a.e. x∈Ω and such that the Jacobian determinant J( x, f) is locally in L 1 log −c 1(n)β L . Then automatically J( x, f) is locally in L 1 log c 2(n)β L(Ω) . This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings.