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- W4225988078 abstract "We prove a single-value version of Reshetnyak's theorem. Namely, if a non-constant map $f in W^{1,n}_{text{loc}}(Omega, mathbb{R}^n)$ from a domain $Omega subset mathbb{R}^n$ satisfies the estimate $lvert Df(x) rvert^n leq K J_f(x) + Sigma(x) lvert f(x) - y_0 rvert^n $ for some $K geq 1$, $y_0in mathbb{R}^n$ and $Sigma in L^{1+varepsilon}_{text{loc}}(Omega)$, then $f^{-1}{y_0}$ is discrete, the local index $i(x, f)$ is positive in $f^{-1}{y_0}$, and every neighborhood of a point of $f^{-1}{y_0}$ is mapped to a neighborhood of $y_0$. Assuming this estimate for a fixed $K$ at every $y_0 in mathbb{R}^n$ is equivalent to assuming that the map $f$ is $K$-quasiregular, even if the choice of $Sigma$ is different for each $y_0$. Since the estimate also yields a single-value Liouville theorem, it hence appears to be a good pointwise definition of $K$-quasiregularity. As a corollary of our single-value Reshetnyak's theorem, we obtain a higher-dimensional version of the argument principle that played a key part in the solution to the Calder'on problem." @default.
- W4225988078 created "2022-05-05" @default.
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- W4225988078 date "2022-02-14" @default.
- W4225988078 modified "2023-10-04" @default.
- W4225988078 title "A single-point Reshetnyak's theorem" @default.
- W4225988078 doi "https://doi.org/10.48550/arxiv.2202.06917" @default.
- W4225988078 hasPublicationYear "2022" @default.
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