{"title":"p-Bilaplacian 的扭转问题","authors":"Andrei Grecu , Mihai Mihăilescu","doi":"10.1016/j.nonrwa.2024.104117","DOIUrl":null,"url":null,"abstract":"<div><p>For each bounded and open set <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span> (<span><math><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></math></span>) with smooth boundary denoted by <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span> and each real number <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span> we analyze the torsion problem of the <span><math><mi>p</mi></math></span>-Bilaplacian, namely <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><msup><mrow><mrow><mo>|</mo><mi>Δ</mi><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>Δ</mi><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> in <span><math><mi>Ω</mi></math></span> with <span><math><mrow><mi>u</mi><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><mn>0</mn></mrow></math></span> on <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span>. Firstly, we show that for each <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span> the problem has a unique weak solution <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>. Secondly, we prove that <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> converges uniformly, as <span><math><mrow><mi>p</mi><mo>→</mo><mi>∞</mi></mrow></math></span>, in <span><math><mrow><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span> to a certain function, say <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, which is exactly the unique solution of the problem <span><math><mrow><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><mn>1</mn></mrow></math></span> in <span><math><mi>Ω</mi></math></span> with <span><math><mrow><mi>u</mi><mo>=</mo><mn>0</mn></mrow></math></span> on <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span>. Moreover, for each real number <span><math><mrow><mi>q</mi><mo>∈</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>Δ</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span> converges strongly to <span><math><mrow><mi>Δ</mi><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> in <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>, as <span><math><mrow><mi>p</mi><mo>→</mo><mi>∞</mi></mrow></math></span>. Next, we show that each solution <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> is also a solution for the minimization problem <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mi>p</mi><mo>;</mo><mi>Ω</mi><mo>)</mo></mrow><mo>≔</mo><munder><mrow><mo>inf</mo></mrow><mrow><mi>u</mi><mo>∈</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>∖</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></munder><mfrac><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mrow><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mrow></mfrac><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><msup><mrow><mrow><mo>|</mo><mi>Δ</mi><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi></mrow></msup><mspace></mspace><mi>d</mi><mi>x</mi></mrow><mrow><msup><mrow><mfenced><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mrow><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mrow></mfrac><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>u</mi><mspace></mspace><mi>d</mi><mi>x</mi></mrow></mfenced></mrow><mrow><mi>p</mi></mrow></msup></mrow></mfrac><mspace></mspace></mrow></math></span> where <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>≔</mo><mrow><mo>{</mo><mi>u</mi><mo>∈</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>∩</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>:</mo><mspace></mspace><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>a</mi><mo>.</mo><mi>e</mi><mo>.</mo><mspace></mspace><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>}</mo></mrow></mrow></math></span>. Further, we show that the function <span><math><mrow><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>∋</mo><mi>p</mi><mo>↦</mo><mi>T</mi><mrow><mo>(</mo><mi>p</mi><mo>;</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> is strictly increasing provided that <span><math><mi>Ω</mi></math></span> is a convex and bounded open set for which <span><math><mrow><msup><mrow><mrow><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mspace></mspace><mi>d</mi><mi>x</mi></mrow></math></span> is small. Finally, using this monotonicity result, we give an alternative variational characterization of the constant <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mi>p</mi><mo>;</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> when <span><math><mrow><msup><mrow><mrow><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mspace></mspace><mi>d</mi><mi>x</mi></mrow></math></span> is small. That last variational characterization fails to hold true when <span><math><mrow><msup><mrow><mrow><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mspace></mspace><mi>d</mi><mi>x</mi><mo>></mo><mn>1</mn></mrow></math></span>.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":null,"pages":null},"PeriodicalIF":1.8000,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The torsion problem of the p-Bilaplacian\",\"authors\":\"Andrei Grecu , Mihai Mihăilescu\",\"doi\":\"10.1016/j.nonrwa.2024.104117\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>For each bounded and open set <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span> (<span><math><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></math></span>) with smooth boundary denoted by <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span> and each real number <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span> we analyze the torsion problem of the <span><math><mi>p</mi></math></span>-Bilaplacian, namely <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><msup><mrow><mrow><mo>|</mo><mi>Δ</mi><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>Δ</mi><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> in <span><math><mi>Ω</mi></math></span> with <span><math><mrow><mi>u</mi><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><mn>0</mn></mrow></math></span> on <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span>. Firstly, we show that for each <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span> the problem has a unique weak solution <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>. Secondly, we prove that <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> converges uniformly, as <span><math><mrow><mi>p</mi><mo>→</mo><mi>∞</mi></mrow></math></span>, in <span><math><mrow><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span> to a certain function, say <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, which is exactly the unique solution of the problem <span><math><mrow><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><mn>1</mn></mrow></math></span> in <span><math><mi>Ω</mi></math></span> with <span><math><mrow><mi>u</mi><mo>=</mo><mn>0</mn></mrow></math></span> on <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span>. Moreover, for each real number <span><math><mrow><mi>q</mi><mo>∈</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>Δ</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span> converges strongly to <span><math><mrow><mi>Δ</mi><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> in <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>, as <span><math><mrow><mi>p</mi><mo>→</mo><mi>∞</mi></mrow></math></span>. Next, we show that each solution <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> is also a solution for the minimization problem <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mi>p</mi><mo>;</mo><mi>Ω</mi><mo>)</mo></mrow><mo>≔</mo><munder><mrow><mo>inf</mo></mrow><mrow><mi>u</mi><mo>∈</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>∖</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></munder><mfrac><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mrow><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mrow></mfrac><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><msup><mrow><mrow><mo>|</mo><mi>Δ</mi><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi></mrow></msup><mspace></mspace><mi>d</mi><mi>x</mi></mrow><mrow><msup><mrow><mfenced><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mrow><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mrow></mfrac><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>u</mi><mspace></mspace><mi>d</mi><mi>x</mi></mrow></mfenced></mrow><mrow><mi>p</mi></mrow></msup></mrow></mfrac><mspace></mspace></mrow></math></span> where <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>≔</mo><mrow><mo>{</mo><mi>u</mi><mo>∈</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>∩</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>:</mo><mspace></mspace><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>a</mi><mo>.</mo><mi>e</mi><mo>.</mo><mspace></mspace><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>}</mo></mrow></mrow></math></span>. Further, we show that the function <span><math><mrow><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>∋</mo><mi>p</mi><mo>↦</mo><mi>T</mi><mrow><mo>(</mo><mi>p</mi><mo>;</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> is strictly increasing provided that <span><math><mi>Ω</mi></math></span> is a convex and bounded open set for which <span><math><mrow><msup><mrow><mrow><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mspace></mspace><mi>d</mi><mi>x</mi></mrow></math></span> is small. Finally, using this monotonicity result, we give an alternative variational characterization of the constant <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mi>p</mi><mo>;</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> when <span><math><mrow><msup><mrow><mrow><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mspace></mspace><mi>d</mi><mi>x</mi></mrow></math></span> is small. That last variational characterization fails to hold true when <span><math><mrow><msup><mrow><mrow><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mspace></mspace><mi>d</mi><mi>x</mi><mo>></mo><mn>1</mn></mrow></math></span>.</p></div>\",\"PeriodicalId\":49745,\"journal\":{\"name\":\"Nonlinear Analysis-Real World Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2024-03-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Analysis-Real World Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1468121824000579\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Analysis-Real World Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1468121824000579","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
对于每个边界光滑的有界开集 Ω⊂RN (N≥2),用 ∂Ω 表示,对于每个实数 p∈(1,∞),我们分析 p-Bilaplacian 的扭转问题,即 Δ(|Δu|p-2Δu)=1 in Ω,u=Δu=0 on ∂Ω。首先,我们证明对于每个 p∈(1,∞),问题都有唯一的弱解 up。其次,我们证明 up 在 C1(Ω¯)中随着 p→∞ 均匀地收敛于某个函数,比如 v2,它正是问题 -Δu=1 in Ω 的唯一解,且 u=0 on ∂Ω。此外,对于每个实数 q∈[1,∞),Δup 在 Lq(Ω)中强收敛于 Δv2,因为 p→∞。接下来,我们证明每个向上的解也是最小化问题 T(p;Ω)≔infu∈Xp(Ω)∖{0}1|Ω|∫Ω|Δu|pdx1|Ω|∫Ωudxp 的解,其中 Xp(Ω)≔{u∈W2,p(Ω)∩W01,p(Ω):u(x)≥0,a.e.x∈Ω} 。此外,我们还证明了函数(1,∞)∋p↦T(p;Ω)是严格递增的,条件是Ω是一个凸的有界开集,且|Ω|-1∫Ωv2dx很小。最后,利用这一单调性结果,我们给出了当|Ω|-1∫Ωv2dx很小时常数T(p;Ω)的另一种变分特征。当|Ω|-1∫Ωv2dx>1时,最后一个变分特性不成立。
For each bounded and open set () with smooth boundary denoted by and each real number we analyze the torsion problem of the -Bilaplacian, namely in with on . Firstly, we show that for each the problem has a unique weak solution . Secondly, we prove that converges uniformly, as , in to a certain function, say , which is exactly the unique solution of the problem in with on . Moreover, for each real number , converges strongly to in , as . Next, we show that each solution is also a solution for the minimization problem where . Further, we show that the function is strictly increasing provided that is a convex and bounded open set for which is small. Finally, using this monotonicity result, we give an alternative variational characterization of the constant when is small. That last variational characterization fails to hold true when .
期刊介绍:
Nonlinear Analysis: Real World Applications welcomes all research articles of the highest quality with special emphasis on applying techniques of nonlinear analysis to model and to treat nonlinear phenomena with which nature confronts us. Coverage of applications includes any branch of science and technology such as solid and fluid mechanics, material science, mathematical biology and chemistry, control theory, and inverse problems.
The aim of Nonlinear Analysis: Real World Applications is to publish articles which are predominantly devoted to employing methods and techniques from analysis, including partial differential equations, functional analysis, dynamical systems and evolution equations, calculus of variations, and bifurcations theory.