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We prove a single-value version of Reshetnyak’s theorem. Namely, if a non-constant map $f\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,n}(\mathrm{\Omega },{\mathbb{R}}^n)$ from a domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ satisfies the estimate $\left|Df\right(x){|}^n\le K{J}_f(x)+\mathrm{\Sigma }(x\left)\right|f\left(x\right)-y_0{|}^n$ at almost every $x\in \mathrm{\Omega }$ for some $K\ge 1$, $y_0\in {\mathbb{R}}^n$ and $\mathrm{\Sigma }\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1+\epsilon }\left(\mathrm{\Omega }\right)$, then $f^{-1}\left\{y_0\right\}$ is discrete, the local index $i(x,f)$ is positive in $f^{-1}\left\{y_0\right\}$, and every neighborhood of a point of $f^{-1}\left\{y_0\right\}$ 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 $\mathrm{\Sigma }$ 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 Calderón problem. 

Abstract The concept of complex harmonic potential in a doubly connected condenser (capacitor) is introduced as an analogue of the real-valued potential of an electrostatic vector field. In this analogy the full differential of a complex potential plays the role of the gradient of the scalar potential in the theory of electrostatics. The main objective in the non-static fields is to rule out having the full differential vanish at some points. Nevertheless, there can be critical points where the Jacobian determinant of the differential turns into zero. The latter is in marked contrast to the case of real-valued potentials. Furthermore, the complex electric capacitor also admits an interpretation of the stored energy intensively studied in the theory of hyperelastic deformations. Engineers interested in electrical systems, such as energy storage devises, might also wish to envision complex capacitors aselectromagnetic condenserswhich, generally, store more energy that the electric capacitors. 

We prove a far-reaching generalization of Rickman’s Picard theorem for a surprisingly large class of mappings, based on the recently developed theory of quasiregular values. Our results are new even in the planar case. 

Abstract We give a full characterization of circle homeomorphisms which admit a homeomorphic extension to the unit disk with finite bi-Sobolev norm. As a special case, a bi-conformal variant of the famous Beurling–Ahlfors extension theorem is obtained. Furthermore we show that the existing extension techniques such as applying either the harmonic or the Beurling–Ahlfors operator work poorly in the degenerated setting. This also gives an affirmative answer to a question of Karafyllia and Ntalampekos.