El Houcine Rami, Elhoussine Azroul, Abdelkrim Barbara
{"title":"Existence of solutions for some quasilinear elliptic system with weight and measure-valued right hand side","authors":"El Houcine Rami, Elhoussine Azroul, Abdelkrim Barbara","doi":"10.1007/s13370-023-01117-w","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span>\\(\\Omega \\)</span> be an open bounded domain in <span>\\(I\\!\\!R^{n},\\)</span> we prove the existence of a solution <i>u</i> for the nonlinear elliptic system </p><div><div><span>$$\\begin{aligned} \\text{(QES) } \\left\\{ \\begin{array}{ll} -div\\sigma \\left( x,u\\left( x\\right) ,Du\\left( x\\right) \\right) = \\mu &{}\\quad \\text{ in } \\Omega \\\\ u = 0 &{}\\quad \\text{ on } \\partial \\Omega , \\end{array} \\right. \\end{aligned}$$</span></div><div>\n (0.1)\n </div></div><p>where <span>\\(\\mu \\)</span> is Radon measure on <span>\\(\\Omega \\)</span> with finite mass. In particular, we show that if the coercivity rate of <span>\\(\\sigma \\)</span> lies in the range <span>\\(]\\frac{s+1}{s},(\\frac{s+1}{s})(2-\\frac{1}{n})]\\)</span> with <span>\\(s\\in \\left( \\frac{n}{p}\\,\\ \\infty \\right) \\cap \\left( \\frac{1}{p-1}\\,\\ \\infty \\right) ,\\)</span> then <i>u</i> is approximately differentiable and the equation holds with <i>Du</i> replaced by <span>\\(\\text{ apDu }\\)</span>. The proof relies on an approximation of <span>\\(\\mu \\)</span> by smooth functions <span>\\(f_{k}\\)</span> and a compactness result for the corresponding solutions <span>\\(u_{k}.\\)</span> This follows from a detailed analysis of the Young measure <span>\\(\\{\\delta _{u}(x)\\otimes \\vartheta (x)\\}\\)</span> generated by the sequence <span>\\({(u_{k},Du_{k})}\\)</span>, and the div-curl type inequality <span>\\(\\langle \\vartheta (x),\\sigma (x,u,\\cdot )\\rangle \\le \\overline{\\sigma }(x)\\langle \\vartheta (x),\\cdot \\rangle \\)</span> for the weak limit <span>\\(\\overline{\\sigma }\\)</span> of the sequence.</p></div>","PeriodicalId":46107,"journal":{"name":"Afrika Matematika","volume":"34 4","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s13370-023-01117-w.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Afrika Matematika","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s13370-023-01117-w","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let \(\Omega \) be an open bounded domain in \(I\!\!R^{n},\) we prove the existence of a solution u for the nonlinear elliptic system
where \(\mu \) is Radon measure on \(\Omega \) with finite mass. In particular, we show that if the coercivity rate of \(\sigma \) lies in the range \(]\frac{s+1}{s},(\frac{s+1}{s})(2-\frac{1}{n})]\) with \(s\in \left( \frac{n}{p}\,\ \infty \right) \cap \left( \frac{1}{p-1}\,\ \infty \right) ,\) then u is approximately differentiable and the equation holds with Du replaced by \(\text{ apDu }\). The proof relies on an approximation of \(\mu \) by smooth functions \(f_{k}\) and a compactness result for the corresponding solutions \(u_{k}.\) This follows from a detailed analysis of the Young measure \(\{\delta _{u}(x)\otimes \vartheta (x)\}\) generated by the sequence \({(u_{k},Du_{k})}\), and the div-curl type inequality \(\langle \vartheta (x),\sigma (x,u,\cdot )\rangle \le \overline{\sigma }(x)\langle \vartheta (x),\cdot \rangle \) for the weak limit \(\overline{\sigma }\) of the sequence.