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Armin Schikorra

Publications and source records attributed to Armin Schikorra.

At least 19 recordsLinked to original sources
arXiv2026

Hölder maps under Pfaffian constraints

Given a one form $λ$ in $\mathbb{R}^N$ and $f: \mathbb{S}^{n} \to \mathbb{R}^N$ with $f^\ast λ= 0$ we discuss the maximal Hölder regularity of extensions $F: \mathbb{B}^{n+1} \to \mathbb{R}^N$ such that $F^\astλ= 0$ in distributional sense. Our analysis applies to the Heisenberg groups $\mathbb{H}_n$. It implies in particular that for all $n \geq 1$ any smooth horizontal map $f: \mathbb{S}^{n} \to \mathbb{H}_n$ can be extended to a $C^α$-map $F: \mathbb{B}^{n+1} \to \mathbb{H}_n$ for some $α> 1/2$. Moreover, if $n \geq 3$ we find $C^α$-embeddings from $\mathbb{B}^{n+1}$ into $\mathbb{H}_n$ for some $α> \frac{1}{2}$. In the appendix we discuss an (as of now unverified) approach to extend these arguments to find $C^α$-embeddings from $\mathbb{B}^{2}$ into $\mathbb{H}_1$ for some $α> \frac{1}{2}$, assisted by GPT-6 Astra.

math.AP↗
arXiv2026

Non-Regularizing properties of $n$-Laplace systems with antisymmetric potentials in critical Lebesgue spaces

For every $n\geq 3$ we construct a bounded discontinuous map $u\in W^{1,n}(\mathbb{B}^n,\mathbb{R}^{n+2})$ solving an $n$-Laplace system with an antisymmetric potential $Ω\in L^n$. This shows that a recent regularity result on $n$-Laplace systems with antisymmetric potentials in Lorentz spaces by the authors is sharp in the sense that we cannot move from Lorentz spaces to classical Lebesgue spaces. This gives in particular a negative answer to a question by Rivière.

math.AP↗
arXiv2026

Non-improvability of sharp endpoint estimates

For an integer $n$ and the parameter $γ\in(0,n)$, the Riesz potential $I_γ$ is known to take boundedly $L^1(\mathbb{R}^n)$ into $L^{\frac{n}{n-γ},\infty}(\mathbb{R}^n)$, and also that the target space is the smallest possible among all rearrangement-invariant Banach function spaces. We study the natural question whether the target space can be improved when the domain space is replaced with a (smaller) Lorentz space $L^{1,q}(\mathbb{R}^n)$ with $q\in(0,1)$. The classical methods cannot be used because the spaces $L^{1,q}(\mathbb{R}^n)$ are not equivalently normable. We develop two new abstract methods, establishing rather general results, a particular consequence of each (albeit achieved through completely different means) being the negative answer to this question. The methods are based on special functional properties of endpoint spaces. The results can be applied to a wide field of operators satisfying certain minimal requirements.

math.FA↗
arXiv2026

On Serrin Interior Regularity Criterion for Navier-Stokes Equations

We revisit Serrin's interior spatial regularity criterion for distributional solutions to the Navier-Stokes equations in $\mathbb R^3$ and considerably relax the hypotheses in two main directions. More precisely, we show that if $u\in{L_t^{s'}L_x^s}$ locally is a distributional solution to the Navier-Stokes equations with $\frac2{s'}+\frac3s=1$ for $s'\in[4,\infty)$, then $u\in L^q_t(C_x^\infty)$ locally for all $q\in(2,s')$. If $s'\in(2,4)$, the same conclusion holds provided that in addition $u\in L_t^4(L_x^p)$ locally, for some $p>1$. In particular, we remove any integrability hypothesis on the vorticity, and we reduce the requirement of integrability in time all the way to $L^4$ from $L^\infty$. To achieve this, we employ a new bootstrap argument, distinct from Serrin's, and we argue that a reduction of the exponent in time integrability does not follow from Serrin's original argument.

math.AP↗
arXiv2026

Minimizing and non-minimizing degree one $W^{s,1/s}$-harmonic maps between spheres

We show that $id:\mathbb{S}^1 \to \mathbb{S}^1$ is \emph{not} a minimizing $W^{s,\frac{1}{s}}$-harmonic map for $s \in (0,\frac{1}{8}$). On the other hand, for $s \in (\frac{1}{3},1)$ it is a local minimizing map, and for $s\in [\frac{1}{2}-\varepsilon,\frac{1}{2}+\varepsilon]$ it is a global minimizer. The usual extension or Fourier techniques being unavailable, our argument relies instead of stability analysis in $s$.

math.AP↗
arXiv2026

On minimizing $W^{s,1/s}$-maps between circles

For $s \in (1/4,1)$ and any degree the only $W^{s,\frac{1}{s}}$-minimizers for $\mathbb{S}^1 \to \mathbb{S}^1$ maps are Blaschke products. This gives a resolution of Open Problems 23 and 24 in Brezis-Mironescu's mappings to the circle book, as well as Brezis' Favorite Open Problem 5.4 in this $s$-range. Previous results of this type were partial and restricted only to a small neighborhood of $s=\frac{1}{2}$. In particular, Brezis' Favorite Open Problem 5.1 is completely settled. Moreover, as a consequence of the argument, one also obtains linearized stability results.

math.AP↗
arXiv2026

A remark on staircase laminates in restricted sets

We slightly extend the convex integration via staircase laminate toolbox recently developed by Kleiner, Müller, Székelyhidi, and Xie. As an example we revisit the proof by Astala-Faraco-Székelyhidi on optimal Meyers' regularity theory via this framework.

math.AP↗
arXiv2026

Existence of nontrival $n$-harmonic maps via min-max methods

For any $n \geq 3$ and any closed manifold $\mathcal{N}$ with $π_{n+k}(\mathcal{N}) \neq \{0\}$ for some $k \geq 0$, we establish the existence of nontrivial $n$-harmonic maps from $\mathbb{S}^n$ into $\mathcal{N}$. When $k\geq 1$, these maps naturally appear as bubbling limits of $p$-harmonic maps with $p > n$, obtained by min-max constructions in the limit $p \to n^+$.

math.AP↗
arXiv2025

Hölder continuous mappings, differential forms and the Heisenberg groups

We develop analysis of Hölder continuous mappings with applications to geometry and topology of the Heisenberg groups. We cover the theory of distributional Jacobians of Hölder continuous mappings and pullbacks of differential forms under Hölder continuous mappings. That includes versions of the change of variables formula and the Stokes theorem for Hölder continuous mappings. The main applications are in the setting of the Heisenberg groups, where we provide a simple proof of a generalization of the Gromov non-embedding theorem, and new results about the Hölder homotopy groups of the Heisenberg groups.

math.DG↗
arXiv2025

Local well-posedness for cubic fractional Schrödinger equations with derivatives on the right-hand side

For $s \in (\frac{1}{2},1]$ we investigate well-posedness of the equation \[ \left ( i \partial_t + (-Δ)^{s} \right ) u = \left (|D|^{1-2s} |u|^2 \right)\ |D|^{2s-1} u \] under small initial data $\|u(0)\|_{H^{\frac{n-2s}{2}}(\mathbb{R}^n)} \ll 1$. This equation is a model equation for for $s$-Schrödinger map equation \[ \partial_t ψ= ψ\wedge (-Δ)^s ψ: \quad ψ: \mathbb{R}^n \times \mathbb{R} \to \mathbb{S}^{2}, \]

math.AP↗
arXiv2024

A note on limiting Calderon-Zygmund theory for transformed $n$-Laplace systems in divergence form

We consider rotated $n$-Laplace systems on the unit ball $B_1 \subset \mathbb{R}^n$ of the form \begin{align*} -\mathrm{div}\left( Q|\nabla u|^{n-2} \nabla u\right) = \mathrm{div}(G), \end{align*} where $u\in W^{1,n}(B_1;\mathbb{R}^N)$, $Q\in W^{1,n}(B_1;SO(N))$ and $G\in L^{\left( \frac{n}{n-1},q \right)}(B_1;\mathbb{R}^n\otimes \mathbb{R}^N)$ for some $0<q<\frac{n}{n-1}$. We prove that $\nabla u\in L^{(n,q(n-1))}_{loc}$ with estimates. As a corollary, we obtain that solutions to $Δ_n u \in \mathcal{H}^1$, where $\mathcal{H}^1$ is the Hardy space, have a higher integrability, namely $\nabla u \in L^{(n,n-1)}_{loc}$.

math.AP↗
arXiv2024

Failure of $L^r$-Calderón-Zygmund estimates for the p-Laplace equation for small $r$

Let $p \neq 2$. For any small enough $r> \max \{p-1,1\}$ and for any $Λ> 1$ there exists a Lipschitz function $u$ and a bounded vectorfield $f$ such that \[ \begin{cases} {\rm div}(|\nabla u|^{p-2} \nabla u) = {\rm div} (f) \quad& \text{in $\mathbb{B}^2$}\\ u=0 &\text{on $\partial \mathbb{B}^2$} \end{cases} \] but \[ \int_{\mathbb{B}^2} |\nabla u|^r \not \leq Λ\int_{\mathbb{B}^2} |f|^{\frac{r}{p-1}}. \] This disproves a conjecture by Iwaniec from 1983. The proof adapts recent convex-integration ideas by Colombo-Tione.

math.AP↗
arXiv2024

A short note on nowhere smooth critical points of polyconvex functionals in arbitrary dimension

For any $M, n \geq 2$ and any open set $Ω\subset \mathbb{R}^n$ we find a smooth, strongly polyconvex function $F\colon \mathbb{R}^{M\times n}\to \mathbb{R}$ and a Lipschitz map $u\colon \mathbb{R}^n \to \mathbb{R}^M$ that is a weak local minimizer of the energy \[ \int_Ω F(Du). \] but with nowhere continuous partial derivatives. This extends celebrated results by Müller-Sverák and Székelyhidi to higher dimensions.

math.AP↗
arXiv2024

On wave systems with antisymmetric potential in dimension d >= 4 and well-posedness for (half-)wave maps

We prove a priori estimates for wave systems of the type \[ \partial_{tt} u - Δu = Ω\cdot du + F(u) \quad \text{in $\mathbb{R}^d \times \mathbb{R}$} \] where $d \geq 4$ and $Ω$ is a suitable antisymmetric potential. We show that the assumptions on $Ω$ are applicable to wave- and half-wave maps, the latter by means of the Krieger-Sire reduction. We thus obtain well-posedness of those equations for small initial data in $\dot{H}^{\frac{d}{2}}(\mathbb{R}^d)$.

math.AP↗
arXiv2024

s-stability for W^{s,n/s}-harmonic maps in homotopy groups

We study $s$-dependence for minimizing $W^{s,n/s}$-harmonic maps $u\colon \mathbb{S}^n \to \mathbb{S}^\ell$ in homotopy classes. Sacks--Uhlenbeck theory shows that, for each $s$, minimizers exist in a generating subset of $π_{n}(\mathbb{S}^\ell)$. We show that this generating subset can be chosen locally constant in $s$. We also show that as $s$ varies the minimal $W^{s,n/s}$-energy in each homotopy class changes continuously. In particular, we provide progress to a question raised by Mironescu and Brezis--Mironescu.

math.AP↗