{"title":"涉及各向异性扩散系数的极奇椭圆方程弱解的连续可微性","authors":"Shuntaro Tsubouchi","doi":"10.1515/acv-2022-0072","DOIUrl":null,"url":null,"abstract":"Abstract In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including the one-Laplacian, and is perturbed by a p -Laplacian-type diffusion operator with <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi mathvariant=\"normal\">∞</m:mi> </m:mrow> </m:math> {1<p<\\infty} . This equation seems analytically difficult to handle near a facet, the place where the gradient vanishes. Our main purpose is to prove that weak solutions are continuously differentiable even across the facet. Here it is of interest to know whether a gradient is continuous when it is truncated near a facet. To answer this affirmatively, we consider an approximation problem, and use standard methods including De Giorgi’s truncation and freezing coefficient methods.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"79 7","pages":"0"},"PeriodicalIF":1.3000,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Continuous differentiability of a weak solution to very singular elliptic equations involving anisotropic diffusivity\",\"authors\":\"Shuntaro Tsubouchi\",\"doi\":\"10.1515/acv-2022-0072\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including the one-Laplacian, and is perturbed by a p -Laplacian-type diffusion operator with <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi mathvariant=\\\"normal\\\">∞</m:mi> </m:mrow> </m:math> {1<p<\\\\infty} . This equation seems analytically difficult to handle near a facet, the place where the gradient vanishes. Our main purpose is to prove that weak solutions are continuously differentiable even across the facet. Here it is of interest to know whether a gradient is continuous when it is truncated near a facet. To answer this affirmatively, we consider an approximation problem, and use standard methods including De Giorgi’s truncation and freezing coefficient methods.\",\"PeriodicalId\":49276,\"journal\":{\"name\":\"Advances in Calculus of Variations\",\"volume\":\"79 7\",\"pages\":\"0\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2023-10-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Calculus of Variations\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1515/acv-2022-0072\",\"RegionNum\":3,\"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":"Advances in Calculus of Variations","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/acv-2022-0072","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Continuous differentiability of a weak solution to very singular elliptic equations involving anisotropic diffusivity
Abstract In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including the one-Laplacian, and is perturbed by a p -Laplacian-type diffusion operator with 1<p<∞ {1
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Advances in Calculus of Variations publishes high quality original research focusing on that part of calculus of variation and related applications which combines tools and methods from partial differential equations with geometrical techniques.
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