关于广义牛顿空气动力学问题

Q2 Mathematics
A. Plakhov
{"title":"关于广义牛顿空气动力学问题","authors":"A. Plakhov","doi":"10.1090/mosc/318","DOIUrl":null,"url":null,"abstract":"<p>We consider the generalized Newton’s least resistance problem for convex bodies: minimize the functional <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-integral Underscript normal upper Omega Endscripts left-parenthesis 1 plus StartAbsoluteValue nabla u left-parenthesis x comma y right-parenthesis EndAbsoluteValue squared right-parenthesis Superscript negative 1 Baseline d x d y\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mo>∬<!-- ∬ --></mml:mo>\n <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>+</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\n <mml:mi>u</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>y</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:msup>\n <mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:msup>\n <mml:mi>d</mml:mi>\n <mml:mi>x</mml:mi>\n <mml:mspace width=\"thinmathspace\" />\n <mml:mi>d</mml:mi>\n <mml:mi>y</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\iint _\\Omega (1 + |\\nabla u(x,y)|^2)^{-1} dx\\, dy</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in the class of concave functions <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u colon normal upper Omega right-arrow left-bracket 0 comma upper M right-bracket\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>u</mml:mi>\n <mml:mo>:<!-- : --></mml:mo>\n <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\n <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mi>M</mml:mi>\n <mml:mo stretchy=\"false\">]</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">u\\colon \\Omega \\to [0,M]</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, where the domain <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega subset-of double-struck upper R squared\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\n <mml:mo>⊂<!-- ⊂ --></mml:mo>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\Omega \\subset \\mathbb {R}^2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is convex and bounded and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M greater-than 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>M</mml:mi>\n <mml:mo>></mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">M > 0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. It has been known (see G. Buttazzo, V. Ferone, and B. Kawohl [Math. Nachr. 173 (1995), pp. 71–89]) that if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u\">\n <mml:semantics>\n <mml:mi>u</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">u</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> solves the problem, then <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue nabla u left-parenthesis x comma y right-parenthesis EndAbsoluteValue greater-than-or-equal-to 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\n <mml:mi>u</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>y</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">|\\nabla u(x,y)| \\ge 1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> at all regular points <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis x comma y right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>y</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(x,y)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> such that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u left-parenthesis x comma y right-parenthesis greater-than upper M\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>u</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>y</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>></mml:mo>\n <mml:mi>M</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">u(x,y) > M</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We prove that if the upper level set <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L equals StartSet left-parenthesis x comma y right-parenthesis colon u left-parenthesis x comma y right-parenthesis equals upper M EndSet\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>L</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>y</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>:<!-- : --></mml:mo>\n <mml:mi>u</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>y</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:mi>M</mml:mi>\n <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">L = \\{ (x,y)\\colon u(x,y) = M \\}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> has nonempty interior, then for almost all points of its boundary <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis x overbar comma y overbar right-parenthesis element-of partial-differential upper L\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mover>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">¯<!-- ¯ --></mml:mo>\n </mml:mover>\n </mml:mrow>\n <mml:mo>,</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mover>\n <mml:mi>y</mml:mi>\n <mml:mo stretchy=\"false\">¯<!-- ¯ --></mml:mo>\n </mml:mover>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi>\n <mml:mi>L</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(\\bar {x}, \\bar {y}) \\in \\partial L</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> one has <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"limit Underscript StartLayout 1st Row left-parenthesis x comma y right-parenthesis right-arrow left-parenthesis x overbar comma y overbar right-parenthesis 2nd Row u left-parenthesis x comma y right-parenthesis greater-than upper M EndLayout Endscripts StartAbsoluteValue nabla u left-parenthesis x comma y right-parenthesis EndAbsoluteValue equals 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:munder>\n <mml:mo movablelimits=\"true\" form=\"prefix\">lim</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mstyle scriptlevel=\"1\">\n <mml:mtable rowspacing=\"0.1em\" columnspacing=\"0em\" displaystyle=\"false\">\n <mml:mtr>\n <mml:mtd>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>y</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mover>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">¯<!-- ¯ --></mml:mo>\n </mml:mover>\n </mml:mrow>\n <mml:mo>,</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mover>\n <mml:mi>y</mml:mi>\n <mml:mo stretchy=\"false\">¯<!-- ¯ --></mml:mo>\n </mml:mover>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mtd>\n </mml:mtr>\n <mml:mtr>\n <mml:mt","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"3 S4","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"On generalized Newton’s aerodynamic problem\",\"authors\":\"A. Plakhov\",\"doi\":\"10.1090/mosc/318\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider the generalized Newton’s least resistance problem for convex bodies: minimize the functional <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-integral Underscript normal upper Omega Endscripts left-parenthesis 1 plus StartAbsoluteValue nabla u left-parenthesis x comma y right-parenthesis EndAbsoluteValue squared right-parenthesis Superscript negative 1 Baseline d x d y\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mo>∬<!-- ∬ --></mml:mo>\\n <mml:mi mathvariant=\\\"normal\\\">Ω<!-- Ω --></mml:mi>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo>+</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo stretchy=\\\"false\\\">|</mml:mo>\\n </mml:mrow>\\n <mml:mi mathvariant=\\\"normal\\\">∇<!-- ∇ --></mml:mi>\\n <mml:mi>u</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>y</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:msup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo stretchy=\\\"false\\\">|</mml:mo>\\n </mml:mrow>\\n <mml:mn>2</mml:mn>\\n </mml:msup>\\n <mml:msup>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:mn>1</mml:mn>\\n </mml:mrow>\\n </mml:msup>\\n <mml:mi>d</mml:mi>\\n <mml:mi>x</mml:mi>\\n <mml:mspace width=\\\"thinmathspace\\\" />\\n <mml:mi>d</mml:mi>\\n <mml:mi>y</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\iint _\\\\Omega (1 + |\\\\nabla u(x,y)|^2)^{-1} dx\\\\, dy</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> in the class of concave functions <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"u colon normal upper Omega right-arrow left-bracket 0 comma upper M right-bracket\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>u</mml:mi>\\n <mml:mo>:<!-- : --></mml:mo>\\n <mml:mi mathvariant=\\\"normal\\\">Ω<!-- Ω --></mml:mi>\\n <mml:mo stretchy=\\\"false\\\">→<!-- → --></mml:mo>\\n <mml:mo stretchy=\\\"false\\\">[</mml:mo>\\n <mml:mn>0</mml:mn>\\n <mml:mo>,</mml:mo>\\n <mml:mi>M</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">]</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">u\\\\colon \\\\Omega \\\\to [0,M]</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, where the domain <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Omega subset-of double-struck upper R squared\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi mathvariant=\\\"normal\\\">Ω<!-- Ω --></mml:mi>\\n <mml:mo>⊂<!-- ⊂ --></mml:mo>\\n <mml:msup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n <mml:mn>2</mml:mn>\\n </mml:msup>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Omega \\\\subset \\\\mathbb {R}^2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is convex and bounded and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper M greater-than 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>M</mml:mi>\\n <mml:mo>></mml:mo>\\n <mml:mn>0</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">M > 0</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. It has been known (see G. Buttazzo, V. Ferone, and B. Kawohl [Math. Nachr. 173 (1995), pp. 71–89]) that if <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"u\\\">\\n <mml:semantics>\\n <mml:mi>u</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">u</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> solves the problem, then <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"StartAbsoluteValue nabla u left-parenthesis x comma y right-parenthesis EndAbsoluteValue greater-than-or-equal-to 1\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo stretchy=\\\"false\\\">|</mml:mo>\\n </mml:mrow>\\n <mml:mi mathvariant=\\\"normal\\\">∇<!-- ∇ --></mml:mi>\\n <mml:mi>u</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>y</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo stretchy=\\\"false\\\">|</mml:mo>\\n </mml:mrow>\\n <mml:mo>≥<!-- ≥ --></mml:mo>\\n <mml:mn>1</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">|\\\\nabla u(x,y)| \\\\ge 1</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> at all regular points <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis x comma y right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>y</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">(x,y)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> such that <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"u left-parenthesis x comma y right-parenthesis greater-than upper M\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>u</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>y</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>></mml:mo>\\n <mml:mi>M</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">u(x,y) > M</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. We prove that if the upper level set <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L equals StartSet left-parenthesis x comma y right-parenthesis colon u left-parenthesis x comma y right-parenthesis equals upper M EndSet\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>L</mml:mi>\\n <mml:mo>=</mml:mo>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>y</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>:<!-- : --></mml:mo>\\n <mml:mi>u</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>y</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>=</mml:mo>\\n <mml:mi>M</mml:mi>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">L = \\\\{ (x,y)\\\\colon u(x,y) = M \\\\}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> has nonempty interior, then for almost all points of its boundary <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis x overbar comma y overbar right-parenthesis element-of partial-differential upper L\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mover>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">¯<!-- ¯ --></mml:mo>\\n </mml:mover>\\n </mml:mrow>\\n <mml:mo>,</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mover>\\n <mml:mi>y</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">¯<!-- ¯ --></mml:mo>\\n </mml:mover>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>∈<!-- ∈ --></mml:mo>\\n <mml:mi mathvariant=\\\"normal\\\">∂<!-- ∂ --></mml:mi>\\n <mml:mi>L</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">(\\\\bar {x}, \\\\bar {y}) \\\\in \\\\partial L</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> one has <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"limit Underscript StartLayout 1st Row left-parenthesis x comma y right-parenthesis right-arrow left-parenthesis x overbar comma y overbar right-parenthesis 2nd Row u left-parenthesis x comma y right-parenthesis greater-than upper M EndLayout Endscripts StartAbsoluteValue nabla u left-parenthesis x comma y right-parenthesis EndAbsoluteValue equals 1\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:munder>\\n <mml:mo movablelimits=\\\"true\\\" form=\\\"prefix\\\">lim</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mstyle scriptlevel=\\\"1\\\">\\n <mml:mtable rowspacing=\\\"0.1em\\\" columnspacing=\\\"0em\\\" displaystyle=\\\"false\\\">\\n <mml:mtr>\\n <mml:mtd>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>y</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo stretchy=\\\"false\\\">→<!-- → --></mml:mo>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mover>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">¯<!-- ¯ --></mml:mo>\\n </mml:mover>\\n </mml:mrow>\\n <mml:mo>,</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mover>\\n <mml:mi>y</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">¯<!-- ¯ --></mml:mo>\\n </mml:mover>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mtd>\\n </mml:mtr>\\n <mml:mtr>\\n <mml:mt\",\"PeriodicalId\":37924,\"journal\":{\"name\":\"Transactions of the Moscow Mathematical Society\",\"volume\":\"3 S4\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-03-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the Moscow Mathematical Society\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/mosc/318\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the Moscow Mathematical Society","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/mosc/318","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 4

摘要

考虑凸体的广义牛顿最小阻力问题:最小化凹函数u类中的泛函∫∫Ω(1 + |)∇u(x,y)| 2) -1 dx dy \iint _ \Omega (1 + | \nabla) u(x,y)|²)^{-1} dx\, dy:Ω→[0,M] u \colon\Omega\to [0,M],其中域Ω∧R 2 \Omega\subset\mathbb R{^2是凸有界的,M > 0 M > 0。这是已知的(见G. Buttazzo, V. Ferone和B. Kawohl[数学])。Nachr. 173 (1995), pp. 71-89]),则|∇u(x,y)|≥1 | }\nabla u(x,y)| \ge 1在所有正则点(x,y) (x,y)使得u(x,y) > M u(x,y) > M。证明如果上水平集L = {(x, y):u(x,y) = M} L = {(x,y) \colon u(x,y) = M}具有非空的内部,那么对于其边界(x¯,y¯)∈∂L (\bar x{,}\bar y{) }\in\partial L有lim (x, y)→(x¯,Y¯)本文章由计算机程序翻译,如有差异,请以英文原文为准。
分享
查看原文 本刊更多论文
On generalized Newton’s aerodynamic problem

We consider the generalized Newton’s least resistance problem for convex bodies: minimize the functional Ω ( 1 + | u ( x , y ) | 2 ) 1 d x d y \iint _\Omega (1 + |\nabla u(x,y)|^2)^{-1} dx\, dy in the class of concave functions u : Ω [ 0 , M ] u\colon \Omega \to [0,M] , where the domain Ω R 2 \Omega \subset \mathbb {R}^2 is convex and bounded and M > 0 M > 0 . It has been known (see G. Buttazzo, V. Ferone, and B. Kawohl [Math. Nachr. 173 (1995), pp. 71–89]) that if u u solves the problem, then | u ( x , y ) | 1 |\nabla u(x,y)| \ge 1 at all regular points ( x , y ) (x,y) such that u ( x , y ) > M u(x,y) > M . We prove that if the upper level set L = { ( x , y ) : u ( x , y ) = M } L = \{ (x,y)\colon u(x,y) = M \} has nonempty interior, then for almost all points of its boundary ( x ¯ , y ¯ ) L (\bar {x}, \bar {y}) \in \partial L one has lim ( x , y ) ( x ¯ , y ¯ )

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society Mathematics-Mathematics (miscellaneous)
自引率
0.00%
发文量
19
期刊介绍: This journal, a translation of Trudy Moskovskogo Matematicheskogo Obshchestva, contains the results of original research in pure mathematics.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信