{"title":"抛物线双相障碍问题","authors":"Siegfried Carl , Patrick Winkert","doi":"10.1016/j.nonrwa.2024.104169","DOIUrl":null,"url":null,"abstract":"<div><p>We prove existence results for the parabolic double phase obstacle problem: Find <span><math><mrow><mi>u</mi><mo>∈</mo><mi>K</mi><mo>⊂</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span> with <span><math><mrow><mi>u</mi><mrow><mo>(</mo><mi>⋅</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span> satisfying <span><span><span><math><mrow><mn>0</mn><mo>∈</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>A</mi><mi>u</mi><mo>+</mo><mi>F</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>∂</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>K</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mspace></mspace><mtext>in</mtext><msubsup><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>∗</mo></mrow></msubsup><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>A</mi><mo>:</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>→</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>∗</mo></mrow></msubsup></mrow></math></span> given by <span><span><span><math><mrow><mi>A</mi><mi>u</mi><mo>≔</mo><mo>−</mo><mo>div</mo><mfenced><mrow><msup><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>+</mo><mi>μ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi></mrow></mfenced><mspace></mspace><mtext>for</mtext><mi>u</mi><mo>∈</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo></mrow></math></span></span></span>is the double phase operator acting on <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>τ</mi><mo>;</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>H</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>H</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> denoting the associated Musielak–Orlicz Sobolev space with generalized homogeneous boundary values. The obstacle is represented by the closed convex set <span><math><mi>K</mi></math></span> with the obstacle function <span><math><mi>ψ</mi></math></span> through <span><span><span><math><mrow><mi>K</mi><mo>=</mo><mrow><mo>{</mo><mi>v</mi><mo>∈</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mspace></mspace><mo>:</mo><mspace></mspace><mi>v</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>≤</mo><mi>ψ</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mtext>for a.a.</mtext><mspace></mspace><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><mi>Q</mi><mo>=</mo><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>τ</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span></span></span>and <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> is the indicator function related to <span><math><mi>K</mi></math></span> with <span><math><mrow><mi>∂</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>K</mi></mrow></msub></mrow></math></span> denoting its subdifferential in the sense of convex analysis.</p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1468121824001093/pdfft?md5=8c45c9bac4bc0752dc7ba149da99cd83&pid=1-s2.0-S1468121824001093-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Parabolic double phase obstacle problems\",\"authors\":\"Siegfried Carl , Patrick Winkert\",\"doi\":\"10.1016/j.nonrwa.2024.104169\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We prove existence results for the parabolic double phase obstacle problem: Find <span><math><mrow><mi>u</mi><mo>∈</mo><mi>K</mi><mo>⊂</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span> with <span><math><mrow><mi>u</mi><mrow><mo>(</mo><mi>⋅</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span> satisfying <span><span><span><math><mrow><mn>0</mn><mo>∈</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>A</mi><mi>u</mi><mo>+</mo><mi>F</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>∂</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>K</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mspace></mspace><mtext>in</mtext><msubsup><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>∗</mo></mrow></msubsup><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>A</mi><mo>:</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>→</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>∗</mo></mrow></msubsup></mrow></math></span> given by <span><span><span><math><mrow><mi>A</mi><mi>u</mi><mo>≔</mo><mo>−</mo><mo>div</mo><mfenced><mrow><msup><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>+</mo><mi>μ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi></mrow></mfenced><mspace></mspace><mtext>for</mtext><mi>u</mi><mo>∈</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo></mrow></math></span></span></span>is the double phase operator acting on <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>τ</mi><mo>;</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>H</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>H</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> denoting the associated Musielak–Orlicz Sobolev space with generalized homogeneous boundary values. The obstacle is represented by the closed convex set <span><math><mi>K</mi></math></span> with the obstacle function <span><math><mi>ψ</mi></math></span> through <span><span><span><math><mrow><mi>K</mi><mo>=</mo><mrow><mo>{</mo><mi>v</mi><mo>∈</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mspace></mspace><mo>:</mo><mspace></mspace><mi>v</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>≤</mo><mi>ψ</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mtext>for a.a.</mtext><mspace></mspace><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><mi>Q</mi><mo>=</mo><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>τ</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span></span></span>and <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> is the indicator function related to <span><math><mi>K</mi></math></span> with <span><math><mrow><mi>∂</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>K</mi></mrow></msub></mrow></math></span> denoting its subdifferential in the sense of convex analysis.</p></div>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-06-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S1468121824001093/pdfft?md5=8c45c9bac4bc0752dc7ba149da99cd83&pid=1-s2.0-S1468121824001093-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1468121824001093\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1468121824001093","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
摘要
我们证明了抛物线双相障碍问题的存在性结果:在 u(⋅,0)=0 时找到 u∈K⊂X0 满足 0∈ut+Au+F(u)+∂IK(u)inX0∗ 的 u∈K⊂X0,其中,A:X0→X0∗,Au≔-div|∇u|p-2∇u+μ(x)|∇u|q-2∇uforu∈X0,是作用于 X0=Lp(0,τ;W01,H(Ω))的双相算子,W01,H(Ω)表示具有广义同质边界值的相关穆西拉克-奥利兹索博廖夫空间。障碍由封闭凸集 K 表示,障碍函数 ψ 通过 K={v∈X0:v(x,t)≤ψ(x,t)for a.a.(x,t)∈Q=Ω×(0,τ)} 表示,IK 是与 K 相关的指示函数,∂IK 表示其在凸分析意义上的次微分。
We prove existence results for the parabolic double phase obstacle problem: Find with satisfying where given by is the double phase operator acting on with denoting the associated Musielak–Orlicz Sobolev space with generalized homogeneous boundary values. The obstacle is represented by the closed convex set with the obstacle function through and is the indicator function related to with denoting its subdifferential in the sense of convex analysis.
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