Regularizing properties of n-Laplace systems with antisymmetric potentials in Lorentz spaces
Regularizing properties of n-Laplace systems with antisymmetric potentials in Lorentz spaces
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洛伦兹空间中具有反对称势的 n-拉普拉斯系统的正则性质
DOI：
10.1007/s00208-023-02727-2
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发表时间：
2023
影响因子：
1.4
通讯作者：
Schikorra, Armin
中科院分区：
文献类型：
--
作者：
Martino, Dorian;Schikorra, Armin
We show continuity of solutionsto the system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\textrm{div} (|\nabla u|^{n-2} \nabla u) = \Omega \cdot |\nabla u|^{n-2} \nabla u \end{aligned}$$\end{document}whenis an-antisymmetric potential – and additionally satisfies a Lorentz-space assumption. To obtain our result we study a rotated n-Laplace system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\textrm{div} (Q|\nabla u|^{n-2} \nabla u) = {\tilde{\Omega }} \cdot |\nabla u|^{n-2} \nabla u, \end{aligned}$$\end{document}whereis the Coulomb gauge which ensures improved Lorentz-space integrability of. Because of the matrix-termQ, 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-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 thatn-harmonic maps into manifolds are continuous if their gradient belongs to the Lorentz-space– which is a trivial and optimal assumption if, and the weakest assumption to date for the regularity of criticaln-harmonic maps, without any added differentiability assumption. We also prove a corresponding result forn-LaplaceH-systems.
We show continuity of solutionsto the system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\textrm{div} (|\nabla u|^{n-2} \nabla u) = \Omega \cdot |\nabla u|^{n-2} \nabla u \end{aligned}$$\end{document}whenis an-antisymmetric potential – and additionally satisfies a Lorentz-space assumption. To obtain our result we study a rotated n-Laplace system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\textrm{div} (Q|\nabla u|^{n-2} \nabla u) = {\tilde{\Omega }} \cdot |\nabla u|^{n-2} \nabla u, \end{aligned}$$\end{document}whereis the Coulomb gauge which ensures improved Lorentz-space integrability of. Because of the matrix-termQ, 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-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 thatn-harmonic maps into manifolds are continuous if their gradient belongs to the Lorentz-space– which is a trivial and optimal assumption if, and the weakest assumption to date for the regularity of criticaln-harmonic maps, without any added differentiability assumption. We also prove a corresponding result forn-LaplaceH-systems.
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