{"title":"BMO 型函数、总变异和 Γ 收敛性","authors":"Panu Lahti, Quoc-Hung Nguyen","doi":"10.1090/proc/16812","DOIUrl":null,"url":null,"abstract":"<p>We study the BMO-type functional <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"kappa Subscript epsilon Baseline left-parenthesis f comma double-struck upper R Superscript n Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>κ</mml:mi> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\kappa _{\\varepsilon }(f,\\mathbb {R}^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which can be used to characterize bounded variation functions <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f element-of normal upper B normal upper V left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant=\"normal\">B</mml:mi> <mml:mi mathvariant=\"normal\">V</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">f\\in \\mathrm {BV}(\\mathbb {R}^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Γ</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-limit of this functional, taken with respect to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript normal l normal o normal c Superscript 1\"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"normal\">l</mml:mi> <mml:mi mathvariant=\"normal\">o</mml:mi> <mml:mi mathvariant=\"normal\">c</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">L^1_{\\mathrm {loc}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-convergence, is known to be <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"one fourth StartAbsoluteValue upper D f EndAbsoluteValue left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\tfrac 14 |Df|(\\mathbb {R}^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Γ</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-limit with respect to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript normal l normal o normal c Superscript normal infinity\"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"normal\">l</mml:mi> <mml:mi mathvariant=\"normal\">o</mml:mi> <mml:mi mathvariant=\"normal\">c</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi mathvariant=\"normal\">∞</mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">L^{\\infty }_{\\mathrm {loc}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-convergence is <disp-formula content-type=\"math/mathml\"> \\[ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"one fourth StartAbsoluteValue upper D Superscript a Baseline f EndAbsoluteValue left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis plus one fourth StartAbsoluteValue upper D Superscript c Baseline f EndAbsoluteValue left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis plus one half StartAbsoluteValue upper D Superscript j Baseline f EndAbsoluteValue left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis comma\"> <mml:semantics> <mml:mrow> <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>a</mml:mi> </mml:msup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>+</mml:mo> <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>c</mml:mi> </mml:msup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>+</mml:mo> <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>j</mml:mi> </mml:msup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\tfrac 14 |D^a f|(\\mathbb {R}^n)+\\tfrac 14 |D^c f|(\\mathbb {R}^n)+\\tfrac 12 |D^j f|(\\mathbb {R}^n),</mml:annotation> </mml:semantics> </mml:math> \\] </disp-formula> which agrees with the “pointwise” limit in the case of special functions of bounded varation.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"32 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"BMO-type functionals, total variation, and Γ-convergence\",\"authors\":\"Panu Lahti, Quoc-Hung Nguyen\",\"doi\":\"10.1090/proc/16812\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study the BMO-type functional <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"kappa Subscript epsilon Baseline left-parenthesis f comma double-struck upper R Superscript n Baseline right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>κ</mml:mi> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\kappa _{\\\\varepsilon }(f,\\\\mathbb {R}^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which can be used to characterize bounded variation functions <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"f element-of normal upper B normal upper V left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant=\\\"normal\\\">B</mml:mi> <mml:mi mathvariant=\\\"normal\\\">V</mml:mi> </mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">f\\\\in \\\\mathrm {BV}(\\\\mathbb {R}^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Gamma\\\"> <mml:semantics> <mml:mi mathvariant=\\\"normal\\\">Γ</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-limit of this functional, taken with respect to <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Subscript normal l normal o normal c Superscript 1\\\"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\\\"normal\\\">l</mml:mi> <mml:mi mathvariant=\\\"normal\\\">o</mml:mi> <mml:mi mathvariant=\\\"normal\\\">c</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding=\\\"application/x-tex\\\">L^1_{\\\\mathrm {loc}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-convergence, is known to be <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"one fourth StartAbsoluteValue upper D f EndAbsoluteValue left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mstyle displaystyle=\\\"false\\\" scriptlevel=\\\"0\\\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mrow> <mml:mo stretchy=\\\"false\\\">|</mml:mo> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo stretchy=\\\"false\\\">|</mml:mo> </mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\tfrac 14 |Df|(\\\\mathbb {R}^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that the <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Gamma\\\"> <mml:semantics> <mml:mi mathvariant=\\\"normal\\\">Γ</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-limit with respect to <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Subscript normal l normal o normal c Superscript normal infinity\\\"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\\\"normal\\\">l</mml:mi> <mml:mi mathvariant=\\\"normal\\\">o</mml:mi> <mml:mi mathvariant=\\\"normal\\\">c</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi mathvariant=\\\"normal\\\">∞</mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding=\\\"application/x-tex\\\">L^{\\\\infty }_{\\\\mathrm {loc}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-convergence is <disp-formula content-type=\\\"math/mathml\\\"> \\\\[ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"one fourth StartAbsoluteValue upper D Superscript a Baseline f EndAbsoluteValue left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis plus one fourth StartAbsoluteValue upper D Superscript c Baseline f EndAbsoluteValue left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis plus one half StartAbsoluteValue upper D Superscript j Baseline f EndAbsoluteValue left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis comma\\\"> <mml:semantics> <mml:mrow> <mml:mstyle displaystyle=\\\"false\\\" scriptlevel=\\\"0\\\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mrow> <mml:mo stretchy=\\\"false\\\">|</mml:mo> </mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>a</mml:mi> </mml:msup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo stretchy=\\\"false\\\">|</mml:mo> </mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>+</mml:mo> <mml:mstyle displaystyle=\\\"false\\\" scriptlevel=\\\"0\\\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mrow> <mml:mo stretchy=\\\"false\\\">|</mml:mo> </mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>c</mml:mi> </mml:msup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo stretchy=\\\"false\\\">|</mml:mo> </mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>+</mml:mo> <mml:mstyle displaystyle=\\\"false\\\" scriptlevel=\\\"0\\\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mrow> <mml:mo stretchy=\\\"false\\\">|</mml:mo> </mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>j</mml:mi> </mml:msup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo stretchy=\\\"false\\\">|</mml:mo> </mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\tfrac 14 |D^a f|(\\\\mathbb {R}^n)+\\\\tfrac 14 |D^c f|(\\\\mathbb {R}^n)+\\\\tfrac 12 |D^j f|(\\\\mathbb {R}^n),</mml:annotation> </mml:semantics> </mml:math> \\\\] </disp-formula> which agrees with the “pointwise” limit in the case of special functions of bounded varation.</p>\",\"PeriodicalId\":20696,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society\",\"volume\":\"32 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-02-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16812\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16812","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们研究了 BMO 型函数 κ ε ( f , R n ) \kappa _{\varepsilon }(f,\mathbb {R}^n),它可以用来描述有界变化函数 f∈ B V ( R n ) f\in \mathrm {BV}(\mathbb {R}^n)。该函数的 Γ \Gamma - Limit 取自 L l o c 1 L^1_{mathrm {loc}} 。 -收敛性,已知为 1 4 | D f | ( R n ) \tfrac 14 |Df|(\mathbb {R}^n) .我们证明,相对于 L l o c ∞ L^{infty }_{mathrm {loc}} 的 Γ \Gamma - Limit 是 -convergence is \[ 1 4 | D a f | ( R n ) + 1 4 | D c f | ( R n ) + 1 2 | D j f | ( R n ) 、 \tfrac 14 |D^a f|(\mathbb {R}^n)+\tfrac 14 |D^c f|(\mathbb {R}^n)+\tfrac 12 |D^j f|(\mathbb {R}^n), \]这与有界变化的特殊函数情况下的 "pointwise "极限一致。
BMO-type functionals, total variation, and Γ-convergence
We study the BMO-type functional κε(f,Rn)\kappa _{\varepsilon }(f,\mathbb {R}^n), which can be used to characterize bounded variation functions f∈BV(Rn)f\in \mathrm {BV}(\mathbb {R}^n). The Γ\Gamma-limit of this functional, taken with respect to Lloc1L^1_{\mathrm {loc}}-convergence, is known to be 14|Df|(Rn)\tfrac 14 |Df|(\mathbb {R}^n). We show that the Γ\Gamma-limit with respect to Lloc∞L^{\infty }_{\mathrm {loc}}-convergence is \[ 14|Daf|(Rn)+14|Dcf|(Rn)+12|Djf|(Rn),\tfrac 14 |D^a f|(\mathbb {R}^n)+\tfrac 14 |D^c f|(\mathbb {R}^n)+\tfrac 12 |D^j f|(\mathbb {R}^n), \] which agrees with the “pointwise” limit in the case of special functions of bounded varation.
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