{"title":"广义黑森商方程的内部\\(C^2\\)估计","authors":"Siyuan Lu","doi":"10.1007/s40818-025-00215-1","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we study the interior <span>\\(C^2\\)</span> regularity problem for the Hessian quotient equation <span>\\(\\left(\\frac{\\sigma_n}{\\sigma_k}\\right)(D^2u)=f\\)</span>. We give a complete answer to this longstanding problem: for <span>\\(k=n-1,n-2\\)</span>, we establish an interior <span>\\(C^2\\)</span> estimate; for <span>\\(k\\leq n-3\\)</span>, we show that interior <span>\\(C^2\\)</span> estimate fails by finding a singular solution.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":"11 2","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Interior \\\\(C^2\\\\) Estimate for Hessian Quotient Equation in General Dimension\",\"authors\":\"Siyuan Lu\",\"doi\":\"10.1007/s40818-025-00215-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we study the interior <span>\\\\(C^2\\\\)</span> regularity problem for the Hessian quotient equation <span>\\\\(\\\\left(\\\\frac{\\\\sigma_n}{\\\\sigma_k}\\\\right)(D^2u)=f\\\\)</span>. We give a complete answer to this longstanding problem: for <span>\\\\(k=n-1,n-2\\\\)</span>, we establish an interior <span>\\\\(C^2\\\\)</span> estimate; for <span>\\\\(k\\\\leq n-3\\\\)</span>, we show that interior <span>\\\\(C^2\\\\)</span> estimate fails by finding a singular solution.</p></div>\",\"PeriodicalId\":36382,\"journal\":{\"name\":\"Annals of Pde\",\"volume\":\"11 2\",\"pages\":\"\"},\"PeriodicalIF\":2.6000,\"publicationDate\":\"2025-06-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Pde\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40818-025-00215-1\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Pde","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s40818-025-00215-1","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Interior \(C^2\) Estimate for Hessian Quotient Equation in General Dimension
In this paper, we study the interior \(C^2\) regularity problem for the Hessian quotient equation \(\left(\frac{\sigma_n}{\sigma_k}\right)(D^2u)=f\). We give a complete answer to this longstanding problem: for \(k=n-1,n-2\), we establish an interior \(C^2\) estimate; for \(k\leq n-3\), we show that interior \(C^2\) estimate fails by finding a singular solution.
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