线性偏差泛函的下半连续性通有失效
The Generic Failure of Lower-semicontinuity for the Linear Distortion Functional
摘要：
我们考虑了欧氏$n$ -空间$n\geq 3$中定义的域的同胚的畸变泛函，特别是线性畸变的凸性。内、外偏差泛函在所有维上都是下半连续的，因此对于曲线模或拟共形的解析定义，可以得出：如果$ \{ f_{n} \}_{n=1}^{\infty} $是$K$ -拟共形映射序列，（这里$K$取决于特定的失真泛函，但是对于序列的每个元素都是相同的），其局部一致地收敛到映射$f$，则这个极限函数也是$K$ -拟共形的。尽管普遍认为这对于拟共形的几何定义（通过线性畸变$H({f_{n}})$定义）也是正确的，T. Iwaniec给出了一个具体的令人惊讶的例子来证明线性偏差泛函在拟共形映射的一致收敛序列上并不总是下连续的。在这里，我们表明，这种故障较低的连续性是常见的，也许是通用的意义上说，在温和的限制下的拟共形$f$，有一个序列$ \{f_{n} \}_{n=1}^{\infty} $与$ {f_{n}}\to {f}$局部一致，并与$\limsup_{n\to\infty} H( {f_{n}})<H( {f})$。我们的主要结果表明，这是真实的仿射映射。解决F.W.的问题。Gehring和Iwaniec证明了极限的跳跃可以是任意大的，并给出了精确的界限：对于每个$α<\sqrt{2}$，有${f_{n}}\to {f}$局部一致地与$f$仿射和\[ α\; \limsup_{n\to\infty} H( {f_{n}}) < H( {f}) \]，我们猜想$\sqrt{2}$是最好的。
英文摘要：
We consider the convexity properties of distortion functionals, particularly the linear distortion, defined for homeomorphisms of domains in Euclidean $n$-spaces, $n\geq 3$. 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}}\to {f}$ locally uniformly and with $\limsup_{n\to\infty} H( {f_{n}})<H( {f})$. Our main result shows this is true for affine mappings. Addressing conjectures of F.W. Gehring and Iwaniec we show the jump up in the limit can be arbitrarily large and give conjecturally sharp bounds : for each $α<\sqrt{2}$ there is ${f_{n}}\to {f}$ locally uniformly with $f$ affine and \[ α\; \limsup_{n\to\infty} H( {f_{n}}) < H( {f}) \] We conjecture $\sqrt{2}$ to be best possible.
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