On generalized Newton’s aerodynamic problem

Q2 Mathematics

Transactions of the Moscow Mathematical Society Pub Date : 2022-03-15 DOI:10.1090/mosc/318

A. Plakhov

{"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</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</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</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":null,"pages":null},"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}

引用次数: 4

Abstract

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 ¯ )

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关于广义牛顿空气动力学问题

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¯)

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