Timeline for Non-injective continuous maps that appear quasiconformal
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 30, 2019 at 11:18 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
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| Apr 5, 2017 at 4:31 | comment | added | Alexandre Eremenko | See, for example, the introduction to this paper Heinonen, Juha, Koskela, Pekka, Definitions of quasiconformality, Invent. Math. 120 (1995), no. 1, 61–79. | |
| Apr 5, 2017 at 4:30 | comment | added | Clark | Ah this is fantastic, this is exactly what I need. What's the best reference for this theorem? | |
| Apr 5, 2017 at 4:27 | vote | accept | Clark | ||
| Apr 5, 2017 at 4:23 | comment | added | Alexandre Eremenko | Yes, but this is a deep theorem. Bounded distortion can be defined on any metric space (+continuity and no other conditions). Such maps are called of bounded distortion. But on R^n they are the same as quasiregular maps (defined with partial derivatives and Sobolev spaces). | |
| Apr 4, 2017 at 23:43 | comment | added | Clark | Thanks! The sticking point for me then is I don't believe I understand the definition of a quasiregular map. For the plane, the definition that I've seen is that $f$ should be in $W^{1,2}_{loc}$ and $\|Df(x)\|^{2} \leq K|J_{f}(x)|$ (in whatever sense that is supposed to mean, it's been a long time since I've studied Sobolev spaces). But I don't know in advance that my function is locally weakly differentiable and in most cases when this bounded distortion property doesn't occur it won't be. Does the bounded distortion property imply that $f \in W^{1,2}_{loc}$? | |
| Apr 4, 2017 at 18:10 | history | edited | Alexandre Eremenko | CC BY-SA 3.0 |
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| Apr 4, 2017 at 18:00 | history | answered | Alexandre Eremenko | CC BY-SA 3.0 |