{"title":"Gradient regularity for widely degenerate elliptic partial differential equations.","authors":"Michael Strunk","doi":"10.1007/s42985-025-00349-8","DOIUrl":null,"url":null,"abstract":"<p><p>In this paper, we investigate the regularity of weak solutions <math><mrow><mi>u</mi> <mo>:</mo> <mi>Ω</mi> <mo>→</mo> <mi>R</mi></mrow> </math> to elliptic equations of the type <dispformula> <math> <mrow> <mtable> <mtr> <mtd><mrow><mtext>div</mtext> <mspace></mspace> <mi>∇</mi> <mi>F</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>D</mi> <mi>u</mi> <mo>)</mo> <mo>=</mo> <mi>f</mi> <mspace></mspace> <mtext>in</mtext> <mspace></mspace> <mi>Ω</mi> <mo>,</mo></mrow> </mtd> </mtr> </mtable> </mrow> </math> </dispformula> whose ellipticity degenerates in a fixed bounded and convex set <math><mrow><mi>E</mi> <mo>⊂</mo> <msup><mrow><mi>R</mi></mrow> <mi>n</mi></msup> </mrow> </math> with <math><mrow><mn>0</mn> <mo>∈</mo> <mtext>Int</mtext> <mspace></mspace> <mi>E</mi></mrow> </math> . Here, <math><mrow><mi>Ω</mi> <mo>⊂</mo> <msup><mrow><mi>R</mi></mrow> <mi>n</mi></msup> </mrow> </math> denotes a bounded domain, and <math><mrow><mi>F</mi> <mo>:</mo> <mi>Ω</mi> <mo>×</mo> <msup><mrow><mi>R</mi></mrow> <mi>n</mi></msup> <mo>→</mo> <msub><mi>R</mi> <mrow><mo>≥</mo> <mn>0</mn></mrow> </msub> </mrow> </math> is a function with the properties: for any <math><mrow><mi>x</mi> <mo>∈</mo> <mi>Ω</mi></mrow> </math> , the mapping <math><mrow><mi>ξ</mi> <mo>↦</mo> <mi>F</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>ξ</mi> <mo>)</mo></mrow> </math> is regular outside <i>E</i> and vanishes entirely within this set. Additionally, we assume <math><mrow><mi>f</mi> <mo>∈</mo> <msup><mi>L</mi> <mrow><mi>n</mi> <mo>+</mo> <mi>σ</mi></mrow> </msup> <mrow><mo>(</mo> <mi>Ω</mi> <mo>)</mo></mrow> </mrow> </math> for some <math><mrow><mi>σ</mi> <mo>></mo> <mn>0</mn></mrow> </math> , representing an arbitrary datum. Our main result establishes the regularity <dispformula> <math> <mrow> <mtable> <mtr> <mtd><mrow><mi>K</mi> <mrow><mo>(</mo> <mi>D</mi> <mi>u</mi> <mo>)</mo></mrow> <mo>∈</mo> <msup><mi>C</mi> <mn>0</mn></msup> <mrow><mo>(</mo> <mi>Ω</mi> <mo>)</mo></mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math> </dispformula> for any continuous function <math><mrow><mi>K</mi> <mo>∈</mo> <msup><mi>C</mi> <mn>0</mn></msup> <mrow><mo>(</mo> <msup><mrow><mi>R</mi></mrow> <mi>n</mi></msup> <mo>)</mo></mrow> </mrow> </math> vanishing on <i>E</i>.</p>","PeriodicalId":74818,"journal":{"name":"SN partial differential equations and applications","volume":"6 5","pages":"39"},"PeriodicalIF":1.6000,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12476422/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SN partial differential equations and applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s42985-025-00349-8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/9/27 0:00:00","PubModel":"Epub","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we investigate the regularity of weak solutions to elliptic equations of the type whose ellipticity degenerates in a fixed bounded and convex set with . Here, denotes a bounded domain, and is a function with the properties: for any , the mapping is regular outside E and vanishes entirely within this set. Additionally, we assume for some , representing an arbitrary datum. Our main result establishes the regularity for any continuous function vanishing on E.