The Generic Failure of Lower-Semicontinuity for the Linear Distortion Functional

Abstract

We consider the convexity properties of distortion functionals, particularly the linear distortion, defined for homeomorphisms of domains in Euclidean n-spaces, \(n\ge 3\). The inner and outer distortion functionals are lower semi-continuous in all dimensions and so for the curve modulus or analytic definitions of quasiconformality it ifollows that if \( \{ f_{n} \}_{n=1}^{\infty } \) is a sequence of K-quasiconformal mappings (here K depends on the particular distortion functional but is the same for every element of the sequence) which converges locally uniformly to a mapping f, then this limit function is also K-quasiconformal. Despite a widespread belief that this was also true for the geometric definition of quasiconformality (defined through the linear distortion \(H({f_{n}})\)), T. Iwaniec gave a specific and surprising example to show that the linear distortion functional is not always lower-semicontinuous on uniformly converging sequences of quasiconformal mappings. Here we show that this failure of lower-semicontinuity is common, perhaps generic in the sense that under mild restrictions on a quasiconformal f, there is a sequence \( \{f_{n} \}_{n=1}^{\infty } \) with \( {f_{n}}\rightarrow {f}\) locally uniformly and with \(\limsup _{n\rightarrow \infty } H( {f_{n}})<H( {f})\). Our main result shows this is true for affine mappings. Addressing conjectures of Gehring and Iwaniec we show the jump up in the limit can be arbitrarily large and give conjecturally sharp bounds: for each \(\alpha <\sqrt{2}\) there is \({f_{n}}\rightarrow {f}\) locally uniformly with f affine and

$$\begin{aligned} \alpha \; \limsup _{n\rightarrow \infty } H( {f_{n}}) < H( {f}) \end{aligned}$$

We conjecture \(\sqrt{2}\) to be best possible.