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Jani Onninen

Publications and source records attributed to Jani Onninen.

At least 19 recordsLinked to original sources
arXiv2026

Values of finite distortion: Reshetnyak's theorem, the Liouville theorem, and the Lusin (N) -property

Let $\Omega \subset \mathbb{R}^n$ be a domain and $f \in W^{1,n}_{\text{loc}} (\Omega,\mathbb{R}^n)$. We say that $f$ has a value of finite distortion at $y_0 \in \mathbb{R}^n$ if there exist measurable functions $K \colon \Omega \to [0,\infty)$ and $\Sigma \in L^1_{\text{loc}} (\Omega)$ such that \[ \lvert Df(x) \rvert^n \le K(x) \det Df (x) + \Sigma(x) \lvert f(x)-y_0 \rvert^n \quad \text{for a.e. } x \in \Omega. \] This notion unifies the classical theory of mappings of finite distortion with the recently introduced theory of quasiregular values. Under sharp integrability assumptions on $K$ and $\Sigma$, we establish single-value analogues of Reshetnyak's theorem and the Liouville theorem. We also prove that mappings satisfying a more general distortion inequality with defect preserve sets of Lebesgue measure zero.

math.CV↗
arXiv2025

Global Integrability of the Reciprocal of Jacobians for Homeomorphisms of Finite Distortion

For a homeomorphism with $p$-integrable distortion, we obtain the optimal global degree of integrability for the reciprocal of its Jacobian determinant. As an application, we strengthen the result of Doležalová, Hencl and Malý concerning weak limits of Sobolev homeomorphisms with finite distortion. Such limits represent physically admissible deformations, as they remain injective almost everywhere and thus adhere as closely as possible to the principle of non-interpenetration of matter in mathematical models of nonlinear elasticity.

math.FA↗
arXiv2025

Continuity for Sobolev mappings with null Lagrangian bounds

We prove the continuity of Sobolev functions $φ\in W^{1,n}_{\mathrm{loc}}(Ω)$, $Ω\subset \mathbb{R}^n$, that satisfy \[ \lvert\nabla φ(x)\rvert^n \le K(x)\bigl(\langle \nabla φ(x), ξ(x)\rangle + A(x)\bigr), \] where $ξ\in L_{\mathrm{loc}}^{n/(n-1)}(Ω, \mathbb{R}^n)$ is weakly divergence-free, and $K \in L^p_{\mathrm{loc}} (Ω)$, $A \in L^q_{\mathrm{loc}} (Ω)$ are non-negative with $p^{-1}+q^{-1}<1$. The result is applicable to a broad class of differential inequalities of null Lagrangian type. As our principal application, we obtain a sharp continuity theorem for $f \in W^{1,n}_{\mathrm{loc}} (Ω, \mathbb{R}^n)$ satisfying the distortion inequality with defect $\lvert Df(x)\rvert^n \le K(x)\det Df(x) + Σ(x)$; this result is new even in the planar case, and closes a significant gap between existing methods and known counterexamples. The proof relies on an overlooked Sobolev-type inequality formulated in terms of measures of superlevel sets.

math.CV↗
arXiv2025

$(INV)$ condition and regularity of the inverse

Let $f \colon Ω\to Ω' $ be a Sobolev mapping of finite distortion between planar domains $Ω$ and $Ω'$, satisfying the $(INV)$ condition and coinciding with a homeomorphism near $\partialΩ$. We show that $f$ admits a generalized inverse mapping $h \colon Ω' \to Ω$, which is also a Sobolev mapping of finite distortion and satisfies the $(INV)$ condition. We also establish a higher-dimensional analogue of this result: if a mapping $f \colon Ω\to Ω' $ of finite distortion is in the Sobolev class $W^{1,p}(Ω, \mathbb{R}^n)$ with $p > n-1$ and satisfies the $(INV)$ condition, then $f$ has an inverse in $W^{1,1}(Ω', \mathbb{R}^n)$ that is also of finite distortion. Furthermore, we characterize Sobolev mappings satisfying $(INV)$ whose generalized inverses have finite $n$-harmonic energy.

math.FA↗
arXiv2025

Characterizing Sobolev Homeomorphic Extensions via Internal Distances

We give a full characterization of embeddings of the unit circle that admit a Sobolev homeomorphic extension to the unit disk. As a direct corollary, we establish that for quasiconvex target domains $\mathbb Y$, any homeomorphism $φ\colon \partial \mathbb{D} \to \partial \mathbb Y$ that admits a continuous $W^{1,p}$-extension to the unit disk $\mathbb{D}$ also admits a $W^{1,p}$-homeomorphic extension. These Sobolev variants of the classical Jordan-Schönflies theorem are essential for ensuring the well-posedness of variational problems arising in Nonlinear Elasticity and Geometric Function Theory.

math.CV↗
arXiv2024

Complex Harmonic Capacitors

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 electrostatic. 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.

math.CV↗
arXiv2024

Linear distortion and rescaling for quasiregular values

Sobolev mappings exhibiting only pointwise quasiregularity-type bounds have arisen in various applications, leading to a recently developed theory of quasiregular values. In this article, we show that by using rescaling, one obtains a direct bridge between this theory and the classical theory of quasiregular maps. More precisely, we prove that a non-constant mapping $f \colon Ω\to \mathbb{R}^n$ with a $(K, Σ)$-quasiregular value at $f(x_0)$ can be rescaled at $x_0$ to a non-constant $K$-quasiregular mapping. Our proof of this fact involves establishing a quasiregular values -version of the linear distortion bound of quasiregular mappings. A quasiregular values variant of the small $K$ -theorem is obtained as an immediate corollary of our main result.

math.CV↗
arXiv2023

Quasiregular values and Rickman's Picard theorem

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.

math.CV↗
arXiv2022

Bi-Sobolev extensions

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.

math.CV↗
arXiv2022

Mappings of generalized finite distortion and continuity

We study continuity properties of Sobolev mappings $f \in W_{\mathrm{loc}}^{1,n} (Ω, \mathbb{R}^n)$, $n \ge 2$, that satisfy the following generalized finite distortion inequality \[\lvert Df(x)\rvert^n \leq K(x) J_f(x) + Σ(x)\] for almost every $x \in \mathbb{R}^n$. Here $K \colon Ω\to [1, \infty)$ and $Σ\colon Ω\to [0, \infty)$ are measurable functions. Note that when $Σ\equiv 0$, we recover the class of mappings of finite distortion, which are always continuous. The continuity of arbitrary solutions, however, turns out to be an intricate question. We fully solve the continuity problem in the case of bounded distortion $K \in L^\infty (Ω)$, where a sharp condition for continuity is that $Σ$ is in the Zygmund space $Σ\log^μ(e + Σ) \in L^1_{\mathrm{loc}}(Ω)$ for some $μ> n-1$. We also show that one can slightly relax the boundedness assumption on $K$ to an exponential class $\exp(λK) \in L^1_{\mathrm{loc}}(Ω)$ with $λ> n+1$, and still obtain continuous solutions when $Σ\log^μ(e + Σ) \in L^1_{\mathrm{loc}}(Ω)$ with $μ> λ$. On the other hand, for all $p, q \in [1, \infty]$ with $p^{-1} + q^{-1} = 1$, we construct a discontinuous solution with $K \in L^p_{\mathrm{loc}}(Ω)$ and $Σ/K \in L^q_{\mathrm{loc}}(Ω)$, including an example with $Σ\in L^\infty_{\mathrm{loc}}(Ω)$ and $K \in L^1_{\mathrm{loc}}(Ω)$.

math.AP↗
arXiv2022

A single-point Reshetnyak's theorem

We prove a single-value version of Reshetnyak's theorem. Namely, if a non-constant map $f \in W^{1,n}_{\text{loc}}(Ω, \mathbb{R}^n)$ from a domain $Ω\subset \mathbb{R}^n$ satisfies the estimate $\lvert Df(x) \rvert^n \leq K J_f(x) + Σ(x) \lvert f(x) - y_0 \rvert^n $ for some $K \geq 1$, $y_0\in \mathbb{R}^n$ and $Σ\in L^{1+\varepsilon}_{\text{loc}}(Ω)$, 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 $Σ$ 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.

math.CV↗
arXiv2022

Fibers of monotone maps of finite distortion

We study topologically monotone surjective $W^{1,n}$-maps of finite distortion $f \colon Ω\to Ω'$, where $Ω, Ω' $ are domains in $\mathbb{R}^n$, $n \geq 2$. If the outer distortion function $K_f \in L_{\mathrm{loc}}^{p}(Ω)$ with $p \geq n-1$, then any such map $f$ is known to be homeomorphic, and hence the fibers $f^{-1}\{y\}$ are singletons. We show that as the exponent of integrability $p$ of the distortion function $K_f$ increases in the range $1/(n-1) \leq p < n-1$, then the fibers $f^{-1}\{y\}$ of $f$ start satisfying increasingly strong homological limitations. We also give a Sobolev realization of a topological example by Bing of a monotone $f \colon \mathbb{R}^3 \to \mathbb{R}^3$ with homologically nontrivial fibers, and show that this example has $K_f \in L^{1/2 - \varepsilon}_{\mathrm{loc}}(\mathbb{R}^3)$ for all $\varepsilon > 0$.

math.AP↗
arXiv2021

Sobolev homeomorphic extensions from two to three dimensions

We study the basic question of characterizing which boundary homeomorphisms of the unit sphere can be extended to a Sobolev homeomorphism of the interior in 3D space. While the planar variants of this problem are well-understood, completely new and direct ways of constructing an extension are required in 3D. We prove, among other things, that a Sobolev homeomorphism $φ\colon \mathbb R^2 \to \mathbb R^2$ in $W_{loc}^{1,p} (\mathbb R^2 , \mathbb R^2)$ for some $p\in [1,\infty )$ admits a homeomorphic extension $h \colon \mathbb R^3 \to \mathbb R^3$ in $W_{loc}^{1,q} (\mathbb R^3, \mathbb R^3) $ for $1\le q < \frac{3}{2}p$. Such an extension result is nearly sharp, as the bound $q=\frac{3}{2}p$ cannot be improved due to the Hölder embedding. The case $q=3$ gains an additional interest as it also provides an $L^1$-variant of the celebrated Beurling-Ahlfors extension result.

math.CA↗
arXiv2021

Analytic characterization of monotone Hopf-harmonics

We study solutions of the inner-variational equation associated with the Dirichlet energy in the plane, given homeomorphic Sobolev boundary data. We prove that such a solution is monotone if and only if its Jacobian determinant does not change sign. These solutions, called monotone Hopf-harmonics, are a natural alternative to harmonic homeomorphisms. Examining the topological behavior of a solution (not a priori monotone) on the trajectories of Hopf quadratic differentials plays a sizable role in our arguments.

math.AP↗
arXiv2021

Energy-minimal Principles in Geometric Function Theory

We survey a number of recent developments in geometric analysis as they pertain to the calculus of variations and extremal problems in geometric function theory following the NZMRI lectures given by the first author at those workshops in Napier in 1998 and 2005.

math.CV↗
arXiv2021

On the heterogeneous distortion inequality

We study Sobolev mappings $f \in W_{\mathrm{loc}}^{1,n} (\mathbb{R}^n, \mathbb{R}^n)$, $n \ge 2$, that satisfy the heterogeneous distortion inequality \[\left|Df(x)\right|^n \leq K J_f(x) + σ^n(x) \left|f(x)\right|^n\] for almost every $x \in \mathbb{R}^n$. Here $K \in [1, \infty)$ is a constant and $σ\geq 0$ is a function in $L^n_{\mathrm{loc}}(\mathbb{R}^n)$. Although we recover the class of $K$-quasiregular mappings when $σ\equiv 0$, the theory of arbitrary solutions is significantly more complicated, partly due to the unavailability of a robust degree theory for non-quasiregular solutions. Nonetheless, we obtain a Liouville-type theorem and the sharp Hölder continuity estimate for all solutions, provided that $σ\in L^{n-\varepsilon}(\mathbb{R}^n) \cap L^{n+\varepsilon}(\mathbb{R}^n)$ for some $\varepsilon >0$. This gives an affirmative answer to a question of Astala, Iwaniec and Martin.

math.CV↗
arXiv2020

The Sobolev Jordan-Schonflies Problem

We consider the planar unit disk $\mathbb D$ as the reference configuration and a Jordan domain $\mathbb Y$ as the deformed configuration, and study the problem of extending a given boundary homeomorphism $φ\colon \partial \mathbb D \to \partial \mathbb Y$ as a Sobolev homeomorphism of the complex plane. Investigating such a Sobolev variant of the classical Jordan-Schönflies theorem is motivated by the well-posedness of the related pure displacement variational questions in the theory of Nonlinear Elasticity (NE) and Geometric Function Theory (GFT). Clearly, the necessary condition for the boundary mapping $φ$ to admit a $W^{1,p}$-Sobolev homeomorphic extension is that it first admits a continuous $W^{1,p}$-Sobolev extension. For an arbitrary target domain $\mathbb Y$ this, however, is not sufficent.

math.CV↗