{"title":"退化Hamilton-Jacobi方程的Beckmann型问题","authors":"Hamza Ennaji, N. Igbida, Van Thanh Nguyen","doi":"10.1090/qam/1606","DOIUrl":null,"url":null,"abstract":"<p>The aim of this note is to give a Beckmann-type problem as well as the corresponding optimal mass transportation problem associated with a degenerate Hamilton-Jacobi equation <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H left-parenthesis x comma nabla u right-parenthesis equals 0 comma\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>H</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\n <mml:mi>u</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">H(x,\\nabla u)=0,</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> coupled with non-zero Dirichlet condition <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u equals g\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>u</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mi>g</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">u=g</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"partial-differential normal upper Omega\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi>\n <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\partial \\Omega</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. In this article, the Hamiltonian <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\n <mml:semantics>\n <mml:mi>H</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is continuous in both arguments, coercive and convex in the second, but not enjoying any property of existence of a smooth strict sub-solution. We also provide numerical examples to validate the correctness of theoretical formulations.</p>","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"Beckmann-type problem for degenerate Hamilton-Jacobi equations\",\"authors\":\"Hamza Ennaji, N. Igbida, Van Thanh Nguyen\",\"doi\":\"10.1090/qam/1606\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The aim of this note is to give a Beckmann-type problem as well as the corresponding optimal mass transportation problem associated with a degenerate Hamilton-Jacobi equation <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper H left-parenthesis x comma nabla u right-parenthesis equals 0 comma\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>H</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi mathvariant=\\\"normal\\\">∇<!-- ∇ --></mml:mi>\\n <mml:mi>u</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>=</mml:mo>\\n <mml:mn>0</mml:mn>\\n <mml:mo>,</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">H(x,\\\\nabla u)=0,</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> coupled with non-zero Dirichlet condition <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"u equals g\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>u</mml:mi>\\n <mml:mo>=</mml:mo>\\n <mml:mi>g</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">u=g</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> on <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"partial-differential normal upper Omega\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi mathvariant=\\\"normal\\\">∂<!-- ∂ --></mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">Ω<!-- Ω --></mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\partial \\\\Omega</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. In this article, the Hamiltonian <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper H\\\">\\n <mml:semantics>\\n <mml:mi>H</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">H</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is continuous in both arguments, coercive and convex in the second, but not enjoying any property of existence of a smooth strict sub-solution. We also provide numerical examples to validate the correctness of theoretical formulations.</p>\",\"PeriodicalId\":20964,\"journal\":{\"name\":\"Quarterly of Applied Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2021-12-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Quarterly of Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/qam/1606\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quarterly of Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/qam/1606","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Beckmann-type problem for degenerate Hamilton-Jacobi equations
The aim of this note is to give a Beckmann-type problem as well as the corresponding optimal mass transportation problem associated with a degenerate Hamilton-Jacobi equation H(x,∇u)=0,H(x,\nabla u)=0, coupled with non-zero Dirichlet condition u=gu=g on ∂Ω\partial \Omega. In this article, the Hamiltonian HH is continuous in both arguments, coercive and convex in the second, but not enjoying any property of existence of a smooth strict sub-solution. We also provide numerical examples to validate the correctness of theoretical formulations.
期刊介绍：
The Quarterly of Applied Mathematics contains original papers in applied mathematics which have a close connection with applications. An author index appears in the last issue of each volume.
This journal, published quarterly by Brown University with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.
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