arXiv subjects

Andreas Gastel

Publications and source records attributed to Andreas Gastel.

7 recordsLinked to original sources

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.

math.AP↗

arXiv2022

Regularity for a geometrically nonlinear flat Cosserat micropolar membrane shell with curvature

We consider the rigorously derived thin shell membrane $Γ$-limit of a three-dimensional isotropic geometrically nonlinear Cosserat micropolar model and deduce full interior regularity of both the midsurface deformation $m:ω\subset{\mathbb R}^2\to{\mathbb R}^3$ and the orthogonal microrotation tensor field $R:ω\subset{\mathbb R}^2\to SO(3)$. The only further structural assumption is that the curvature energy depends solely on the uni-constant isotropic Dirichlet type energy term $|DR|^2$. We use Rivière's regularity techniques of harmonic map type systems for our system which couples harmonic maps to $SO(3)$ with a linear equation for $m$. The additional coupling term in the harmonic map equation is of critical integrability and can only be handled because of its special structure.

math.AP↗

arXiv2019

Conservation laws for even order systems of polyharmonic map type

Following Rivière's study of conservation laws for second order quasilinear systems with critical nonlinearty and Lamm/Rivière's generalization to fourth order, we consider similar systems of order $2m$. Typical examples are $m$-polyharmonic maps. Under natural conditions, we find a conservation law for weak solutions on $2m$-dimensional domains. This implies continuity of weak solutions.

math.AP↗

arXiv2018

Canonical gauges in higher gauge theory

We study the problem of finding good gauges for connections in higher gauge theories. We find that, for $2$-connections in strict $2$-gauge theory and $3$-connections in $3$-gauge theory, there are local "Coulomb gauges" that are more canonical than in classical gauge theory. In particular, they are essentially unique, and no smallness of curvature is needed in the critical dimensions. We give natural definitions of $2$-Yang-Mills and $3$-Yang-Mills theory and find that the choice of good gauges makes them essentially linear. As an application, (anti-)selfdual $2$-connections over $B^6$ are always $2$-Yang-Mills, and (anti-)selfdual $3$-connections over $B^8$ are always $3$-Yang-Mills.

math-ph↗

arXiv2017

Regularity issues for Cosserat continua and $p$-harmonic maps

For minimizers in a geometrically nonlinear Cosserat model for micropolar elasticity of continua, we prove interior Hölder regularity, up to isolated singular points that may be possible if the exponent $p$ from the model is $2$ or in $(\frac{32}{15},3)$. The obstacle to full continuity turns out to be the existence of certain minimizing homogeneous $p$-harmonic maps to $S^3$. For those, we slightly improve existing regularity theorems in order to achieve our result on the Cosserat model.

math.AP↗

arXiv2015

Minimizers of higher order gauge invariant functionals

We introduce higher order variants of the Yang-Mills functional that involve $(n-2)$th order derivatives of the curvature. We prove coercivity and smoothness of critical points in Uhlenbeck gauge in dimensions $\mathrm{dim}M\le 2n$. These results are then used to establish the existence of smooth minimizers on a given principal bundle $P\to M$ for subcritical dimensions $\mathrm{dim}M<2n$. In the case of critical dimension $\mathrm{dim}M=2n$ we construct a minimizer on a bundle which might differ from the prescribed one, but has the same Chern classes $c_1,\ldots,c_{n-1}$. A key result is a removable singularity theorem for bundles carrying a $W^{n-1,2}$-connection. This generalizes a recent result by Petrache and Rivière.

math.AP↗

arXiv2011

Yang-Mills connections of cohomogeneity one on SO(n)-bundles over Euclidean spheres

We construct Yang-Mills connections on SO(n)-bundles over spheres equipped with the Euclidean metric. We use a cohomogeneity one group action on the bundle to reduce the Yang-Mills-equation to a system of ordinary differential equations. The system is shown to have solutions by variational methods, using ideas from harmonic map theory. Examples include Yang-Mills connections on each of the countably many principal SO(6)-bundles over $S^6$, and countably many Yang-Mills connections on $TS^n$ for $n\in\{5,...,9\}$.

math.DG↗