{"title":"On the improvement of Hölder seminorms in superquadratic Hamilton-Jacobi equations","authors":"Marco Cirant","doi":"10.1016/j.jfa.2024.110692","DOIUrl":null,"url":null,"abstract":"<div><div>We show in this paper that maximal <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span>-regularity for time-dependent viscous Hamilton-Jacobi equations with unbounded right-hand side and superquadratic <em>γ</em>-growth in the gradient holds in the full range <span><math><mi>q</mi><mo>></mo><mo>(</mo><mi>N</mi><mo>+</mo><mn>2</mn><mo>)</mo><mfrac><mrow><mi>γ</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>γ</mi></mrow></mfrac></math></span>. Our approach is based on new <span><math><mfrac><mrow><mi>γ</mi><mo>−</mo><mn>2</mn></mrow><mrow><mi>γ</mi><mo>−</mo><mn>1</mn></mrow></mfrac></math></span>-Hölder estimates, which are consequence of the decay at small scales of suitable nonlinear space and time Hölder quotients. This is obtained by proving suitable oscillation estimates, that also give in turn some Liouville type results for entire solutions.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7000,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S002212362400380X/pdfft?md5=9f67759f78f3d63a96e6edeef4bf4034&pid=1-s2.0-S002212362400380X-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S002212362400380X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We show in this paper that maximal -regularity for time-dependent viscous Hamilton-Jacobi equations with unbounded right-hand side and superquadratic γ-growth in the gradient holds in the full range . Our approach is based on new -Hölder estimates, which are consequence of the decay at small scales of suitable nonlinear space and time Hölder quotients. This is obtained by proving suitable oscillation estimates, that also give in turn some Liouville type results for entire solutions.
期刊介绍：
The Journal of Functional Analysis presents original research papers in all scientific disciplines in which modern functional analysis plays a basic role. Articles by scientists in a variety of interdisciplinary areas are published.
Research Areas Include:
• Significant applications of functional analysis, including those to other areas of mathematics
• New developments in functional analysis
• Contributions to important problems in and challenges to functional analysis
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