F. Mamedov, S. Monsurrò
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{"title":"具有非一致退化梯度的Sobolev不等式","authors":"F. Mamedov, S. Monsurrò","doi":"10.14232/ejqtde.2022.1.24","DOIUrl":null,"url":null,"abstract":"<jats:p>In this paper we prove the following weighted Sobolev inequality in a bounded domain <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math>, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:math>, of a homogeneous space <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:math>, under suitable compatibility conditions on the positive weight functions <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> </mml:math> and on the quasi-metric <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ρ<!-- ρ --></mml:mi> </mml:math>, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo maxsize=\"1.623em\" minsize=\"1.623em\">(</mml:mo> </mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> </mml:msub> <mml:mo fence=\"false\" stretchy=\"false\">|</mml:mo> <mml:mi>f</mml:mi> <mml:msup> <mml:mo fence=\"false\" stretchy=\"false\">|</mml:mo> <mml:mi>q</mml:mi> </mml:msup> <mml:mi>v</mml:mi> <mml:mspace width=\"thinmathspace\" /> <mml:mi>w</mml:mi> <mml:mi>d</mml:mi> <mml:mi>z</mml:mi> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo maxsize=\"1.623em\" minsize=\"1.623em\">)</mml:mo> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>q</mml:mi> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>C</mml:mi> </mml:mrow> <mspace width=\"thinmathspace\" /> <mml:munderover> <mml:mo movablelimits=\"false\">∑<!-- ∑ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>N</mml:mi> </mml:munderover> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo maxsize=\"1.623em\" minsize=\"1.623em\">(</mml:mo> </mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> </mml:msub> <mml:mo fence=\"false\" stretchy=\"false\">|</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:msup> <mml:mo fence=\"false\" stretchy=\"false\">|</mml:mo> <mml:mi>p</mml:mi> </mml:msup> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>S</mml:mi> </mml:msub> <mml:mi>w</mml:mi> <mml:mspace width=\"thinmathspace\" /> <mml:mi>d</mml:mi> <mml:mi>z</mml:mi> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo maxsize=\"1.623em\" minsize=\"1.623em\">)</mml:mo> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>p</mml:mi> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mspace width=\"1em\" /> <mml:mi>f</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">L</mml:mi> <mml:mi mathvariant=\"normal\">i</mml:mi> <mml:mi mathvariant=\"normal\">p</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mover> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> <mml:mo accent=\"false\">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>,</mml:mo> </mml:math> where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>q</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>S</mml:mi> </mml:msub> </mml:math> denotes the strong maximal operator. Some corollaries on non-uniformly degenerating gradient inequalities are derived.</jats:p>","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Sobolev inequality with non-uniformly degenerating gradient\",\"authors\":\"F. Mamedov, S. Monsurrò\",\"doi\":\"10.14232/ejqtde.2022.1.24\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<jats:p>In this paper we prove the following weighted Sobolev inequality in a bounded domain <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi mathvariant=\\\"normal\\\">Ω<!-- Ω --></mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math>, <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:math>, of a homogeneous space <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:math>, under suitable compatibility conditions on the positive weight functions <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:math> and on the quasi-metric <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>ρ<!-- ρ --></mml:mi> </mml:math>, <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"block\\\"> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mo maxsize=\\\"1.623em\\\" minsize=\\\"1.623em\\\">(</mml:mo> </mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi mathvariant=\\\"normal\\\">Ω<!-- Ω --></mml:mi> </mml:msub> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">|</mml:mo> <mml:mi>f</mml:mi> <mml:msup> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">|</mml:mo> <mml:mi>q</mml:mi> </mml:msup> <mml:mi>v</mml:mi> <mml:mspace width=\\\"thinmathspace\\\" /> <mml:mi>w</mml:mi> <mml:mi>d</mml:mi> <mml:mi>z</mml:mi> <mml:msup> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mo maxsize=\\\"1.623em\\\" minsize=\\\"1.623em\\\">)</mml:mo> </mml:mrow> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>q</mml:mi> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi>C</mml:mi> </mml:mrow> <mspace width=\\\"thinmathspace\\\" /> <mml:munderover> <mml:mo movablelimits=\\\"false\\\">∑<!-- ∑ --></mml:mo> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>N</mml:mi> </mml:munderover> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mo maxsize=\\\"1.623em\\\" minsize=\\\"1.623em\\\">(</mml:mo> </mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi mathvariant=\\\"normal\\\">Ω<!-- Ω --></mml:mi> </mml:msub> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">|</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:msup> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">|</mml:mo> <mml:mi>p</mml:mi> </mml:msup> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>S</mml:mi> </mml:msub> <mml:mi>w</mml:mi> <mml:mspace width=\\\"thinmathspace\\\" /> <mml:mi>d</mml:mi> <mml:mi>z</mml:mi> <mml:msup> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mo maxsize=\\\"1.623em\\\" minsize=\\\"1.623em\\\">)</mml:mo> </mml:mrow> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>p</mml:mi> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mspace width=\\\"1em\\\" /> <mml:mi>f</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi mathvariant=\\\"normal\\\">L</mml:mi> <mml:mi mathvariant=\\\"normal\\\">i</mml:mi> <mml:mi mathvariant=\\\"normal\\\">p</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mover> <mml:mi mathvariant=\\\"normal\\\">Ω<!-- Ω --></mml:mi> <mml:mo accent=\\\"false\\\">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>,</mml:mo> </mml:math> where <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>q</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:math> and <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>S</mml:mi> </mml:msub> </mml:math> denotes the strong maximal operator. Some corollaries on non-uniformly degenerating gradient inequalities are derived.</jats:p>\",\"PeriodicalId\":50537,\"journal\":{\"name\":\"Electronic Journal of Qualitative Theory of Differential Equations\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2022-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Electronic Journal of Qualitative Theory of Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.14232/ejqtde.2022.1.24\",\"RegionNum\":4,\"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":"Electronic Journal of Qualitative Theory of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.14232/ejqtde.2022.1.24","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Sobolev inequality with non-uniformly degenerating gradient
In this paper we prove the following weighted Sobolev inequality in a bounded domain Ω ⊂ R n , n ≥ 1 , of a homogeneous space ( R n , ρ , w d x ) , under suitable compatibility conditions on the positive weight functions ( v , w , ω 1 , ω 2 , … , ω n ) and on the quasi-metric ρ , ( ∫ Ω | f | q v w d z ) 1 q ≤ C ∑ i = 1 N ( ∫ Ω | f z i | p ω i M S w d z ) 1 p , f ∈ L i p 0 ( Ω ¯ ) , where q ≥ p > 1 and M S denotes the strong maximal operator. Some corollaries on non-uniformly degenerating gradient inequalities are derived.
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