{"title":"Conservation Laws for p-Harmonic Systems with Antisymmetric Potentials and Applications","authors":"Francesca Da Lio, Tristan Rivière","doi":"10.1007/s00205-025-02085-0","DOIUrl":null,"url":null,"abstract":"<div><p>We prove that <i>p</i>-harmonic systems with antisymmetric potentials of the form </p><div><div><span>$$\\begin{aligned} -\\,\\text{ div }\\left( (1+|\\nabla u|^2)^{\\frac{p}{2}-1}\\,\\nabla u\\right) =(1+|\\nabla u|^2)^{\\frac{p}{2}-1}\\,\\Omega \\cdot \\nabla u, \\end{aligned}$$</span></div></div><p>(<span>\\(\\Omega \\)</span> is antisymmetric) can be written in divergence form as a conservation law </p><div><div><span>$$\\begin{aligned} -\\text{ div }\\left( (1+|\\nabla u|^2)^{\\frac{p}{2}-1}\\,A\\,\\nabla u\\right) =\\nabla ^\\perp B\\cdot \\nabla u. \\end{aligned}$$</span></div></div><p>This extends to the <i>p</i>-harmonic framework the original work of the second author for <span>\\(p=2\\)</span> (see Rivière in Invent Math 168(1):1–22, 2007). We give applications of the existence of this divergence structure in the analysis <span>\\(p\\rightarrow 2\\)</span>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 2","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-025-02085-0.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Rational Mechanics and Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00205-025-02085-0","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that p-harmonic systems with antisymmetric potentials of the form
$$\begin{aligned} -\,\text{ div }\left( (1+|\nabla u|^2)^{\frac{p}{2}-1}\,\nabla u\right) =(1+|\nabla u|^2)^{\frac{p}{2}-1}\,\Omega \cdot \nabla u, \end{aligned}$$
(\(\Omega \) is antisymmetric) can be written in divergence form as a conservation law
$$\begin{aligned} -\text{ div }\left( (1+|\nabla u|^2)^{\frac{p}{2}-1}\,A\,\nabla u\right) =\nabla ^\perp B\cdot \nabla u. \end{aligned}$$
This extends to the p-harmonic framework the original work of the second author for \(p=2\) (see Rivière in Invent Math 168(1):1–22, 2007). We give applications of the existence of this divergence structure in the analysis \(p\rightarrow 2\).
期刊介绍:
The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.