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1.
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 ΩRn, n2, and such that exp(βK(x))Lloc1(Ω), β>0. We show that there exist two universal constants c1(n),c2(n) with the following property: Let f be a mapping in Wloc1,1(Ω,Rn) with |Df(x)|nK(x)J(x,f) for a.e. xΩ and such that the Jacobian determinant J(x,f) is locally in L1 log−c1(n)βL. Then automatically J(x,f) is locally in L1 logc2(n)βL(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. 相似文献
2.
Let D(G) be the minimum quantifier depth of a first order sentence Φ that defines a graph G up to isomorphism. Let D0(G) be the version of D(G) where we do not allow quantifier alternations in Φ. Define q0(n) to be the minimum of D0(G) over all graphs G of order n.We prove that for all n we have
log*n−log*log*n−2≤q0(n)≤log*n+22,