تقرير
Regularizing properties of n-Laplace systems with antisymmetric potentials in Lorentz spaces
| العنوان: | Regularizing properties of n-Laplace systems with antisymmetric potentials in Lorentz spaces |
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| المؤلفون: | Martino, Dorian, Schikorra, Armin |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs |
| الوصف: | We show continuity of solutions $u \in W^{1,n}(B^n,\mathbb{R}^N)$ to the system \[ -{\rm div} (|\nabla u|^{n-2} \nabla u) = \Omega \cdot |\nabla u|^{n-2} \nabla u \] when $\Omega$ is an $L^n$-antisymmetric potential -- and additionally satisfies a Lorentz-space assumption. To obtain our result we study a rotated n-Laplace system \[ -{\rm div} (Q|\nabla u|^{n-2} \nabla u) = \tilde{\Omega} \cdot |\nabla u|^{n-2} \nabla u, \] where $Q \in W^{1,n}(B^n,SO(N))$ is the Coulomb gauge which ensures improved Lorentz-space integrability of $\tilde{\Omega}$. Because of the matrix-term $Q$, this system does not fall directly into Kuusi-Mingione's vectorial potential theory. However, we adapt ideas of their theory together with Iwaniec' stability result to obtain $L^{(n,\infty)}$-estimates of the gradient of a solution which, by an iteration argument leads to the regularity of solutions. As a corollary of our argument we see that $n$-harmonic maps into manifolds are continuous if their gradient belongs to the Lorentz-space $L^{(n,2)}$ -- which is a trivial and optimal assumption if $n=2$, and the weakest assumption to date for the regularity of critical $n$-harmonic maps, without any added differentiability assumption. We also discuss an application to H systems. Comment: Added application to H-systems |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.1007/s00208-023-02727-2 |
| URL الوصول: | http://arxiv.org/abs/2305.05961 |
| رقم الأكسشن: | edsarx.2305.05961 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.1007/s00208-023-02727-2 |
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