arXiv subjects

Michał Miśkiewicz

Publications and source records attributed to Michał Miśkiewicz.

9 recordsLinked to original sources

arXiv2026

Regularity of minimizing $p$-harmonic maps from $B^3$ to $\mathbb{S}^3$

We prove that every minimizing $p$-harmonic map from $B^3$ into $\mathbb S^3$ is locally $C^{1,α}$ for some $α\in(0,1)$, for every $p>p_0$, where $p_0=\frac{7-\sqrt{17}}{2}\approx 1.44$. This closes the gap between the previously established regularity ranges $[2,2.642]\cup[2.961,3]$ and, in particular, yields full interior regularity for all $p\ge2$. The result also extends regularity to the subquadratic range $p_0<p<2$.

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arXiv2025

Vectorial Kato inequality for $p$-harmonic maps with optimal constant

We derive the sharp vectorial Kato inequality for $p$-harmonic mappings. Surprisingly, the optimal constant differs from the one obtained for scalar valued $p$-harmonic functions by Chang, Chen, and Wei. As an application we demonstrate how this inequality can be used in the study of regularity of $p$-harmonic maps. Furthermore, in the case of $p$-harmonic maps from $B^3$ to $\mathbb{S}^3$, we enhance the known range of $p$ values for which regularity is achieved. Specifically, we establish that for $p \in [2, 2.642]$, minimizing $p$-harmonic maps must be regular.

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arXiv2024

Regularity of minimizing $p$-harmonic maps into spheres and sharp Kato inequality

We study regularity of minimizing $p$-harmonic maps $u \colon B^3 \to \mathbb{S}^3$ for $p$ in the interval $[2,3]$. For a long time, regularity was known only for $p = 3$ (essentially due to Morrey) and $p = 2$ (Schoen-Uhlenbeck), but recently Gastel extended the latter result to $p \in [2,2+\frac{2}{15}]$ using a version of Kato inequality. Here, we establish regularity for a small interval $p\in [2.961,3]$ by combining Morrey's methods with Hardt and Lin's Extension Theorem. We also improve on the other result by obtaining regularity for $p \in [2,p_0]$ with $p_0 = \frac{3+\sqrt{3}}{2} \approx 2.366$. In relation to this, we address a question posed by Gastel and prove a sharp Kato inequality for $p$-harmonic maps in two-dimensional domains, which is of independent interest.

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arXiv2023

A new $p$-harmonic map flow with Struwe monotonicity

We construct and analyze solutions to a regularized homogeneous $p$-harmonic map flow equation for general $p \geq 2$. The homogeneous version of the problem is new and features a monotonicity formula extending the one found by Struwe for $p = 2$; such a formula is not available for the nonhomogeneous equation. The construction itself is via a Ginzburg-Landau-type approximation à la Chen-Struwe, employing tools such as a Bochner-type formula and an $\varepsilon$-regularity theorem. We similarly obtain strong subsequential convergence of the approximations away from a concentration set with parabolic codimension at least $p$. However, the quasilinear and non-divergence nature of the equation presents new obstacles that do not appear in the classical case $p = 2$, namely uniform-time existence for the approximating problem, and thus our basic existence result is stated conditionally.

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arXiv2022

Regularity for solutions of H-systems and n-harmonic maps with n/2 square integrable derivatives

We study the regularity of weak solutions for two elliptic systems involving the $n$-Laplacian and a critical nonlinearity in the right hand side: $H$-systems and $n$-harmonic maps into compact Riemannian manifolds. Under the assumptions that the solutions belong to $W^{n/2,2}$ in an even dimension $n$, we prove their continuty. The tools used in the proof involve Hardy spaces and BMO, and the Rivière--Uhlenbeck decomposition (with estimates in Morrey spaces). A prominent role is played by the Coifman--Rochberg--Weiss commutator theorem.

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arXiv2021

On the size of the singular set of minimizing harmonic maps

We consider minimizing harmonic maps $u$ from $Ω\subset \mathbb{R}^n$ into a closed Riemannian manifold $\mathcal{N}$ and prove: (1) an extension to $n \geq 4$ of Almgren and Lieb's linear law. That is, if the fundamental group of the target manifold $\mathcal{N}$ is finite, we have \[ \mathcal{H}^{n-3}(\textrm{sing } u) \le C \int_{\partial Ω} |\nabla_T u|^{n-1} \,d \mathcal{H}^{n-1}; \] (2) an extension of Hardt and Lin's stability theorem. Namely, assuming that the target manifold is $\mathcal{N}=\mathbb{S}^2$ we obtain that the singular set of $u$ is stable under small $W^{1,n-1}$-perturbations of the boundary data. In dimension $n=3$ both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$ with $s \in (\frac{1}{2},1]$ and $p \in [2,\infty)$ satisfying $sp \geq 2$. We also discuss sharpness.

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arXiv2019

On the size of the singular set of minimizing harmonic maps into the 2-sphere in dimension four and higher

We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions $n \geq 4$. For minimizing harmonic maps $u\in W^{1,2}(Ω,\mathbb{S}^2)$ from $n$-dimensional domains into the two dimensional sphere we prove: (1) An extension of Almgren and Lieb's linear law, namely \[\mathcal{H}^{n-3}(\textrm{sing} u) \le C \int_{\partial Ω} |\nabla_T u|^{n-1} \,d\mathcal{H}^{n-1};\] (2) An extension of Hardt and Lin's stability theorem, namely that the size of singular set is stable under small perturbations in $W^{1,n-1}$ norm of the boundary.

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arXiv2018

On Hölder regularity of the singular set of energy minimizing harmonic maps into closed manifolds

Energy minimizing harmonic maps between manifolds are known to be smooth outside a rectifiable set of codimension $3$, called the singular set. The possibility that this set is not a manifold, but has arbitrarily many small gaps in it, is not excluded in general. Here we prove that some part of the singular set - characterized by topological and analytic properties of tangent maps - is a topological manifold. In the special case of maps into the sphere ${\mathbb S}^2$, we conclude that the whole top-dimensional part of the singular set is a manifold - this generalizes a similar result in two-dimensional domain, due to Hardt and Lin.

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arXiv2017

Fractional differentiability for solutions of the inhomogenous $p$-Laplace system

It is shown that if $p \ge 3$ and $u \in W^{1,p}(Ω,\mathbb{R}^N)$ solves the inhomogenous $p$-Laplace system \[ \operatorname{div} (|\nabla u|^{p-2} \nabla u) = f, \qquad f \in W^{1,p'}(Ω,\mathbb{R}^N), \] then locally the gradient $\nabla u$ lies in the fractional Nikol'skii space $\mathcal{N}^{θ,2/θ}$ with any $θ\in [ \tfrac{2}{p}, \tfrac{2}{p-1} )$. To the author's knowledge, this result is new even in the case of $p$-harmonic functions, slightly improving known $\mathcal{N}^{2/p,p}$ estimates. The method used here is an extension of the one used by A. Cellina in the case $2 \le p < 3$ to show $W^{1,2}$ regularity.

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