Weighted \(L^p\) -theory for Poisson, biharmonic and Stokes problems on periodic unbounded strips of 详细信息    查看全文

• 作者：Vuk Milišić ; Ulrich Razafison
• 刊名：Annali dell'Universita di Ferrara
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：62
• 期：1
• 页码：117-142
• 全文大小：583 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Analysis
  Geometry
  History of Mathematics
  Numerical Analysis
  Algebraic Geometry
  
• 出版者：Springer Milan
• ISSN：1827-1510
• 卷排序：62

文摘

This paper establishes isomorphisms for Laplace, biharmonic and Stokes operators in weighted Sobolev spaces. The \(W^{m,p}_{\alpha }({{\mathbb {R}}}^n)\)-spaces are similar to standard Sobolev spaces \(W^{m,p}_{}({\mathbb {R}}^n)\), but they are endowed with weights \((1+|x|^2)^{\alpha /2}\) prescribing functions’ growth or decay at infinity. Although well established in \({{\mathbb {R}}}^n\) [3], these weighted results do not apply in the specific hypothesis of periodicity. This kind of problem appears when studying singularly perturbed domains (roughness, sieves, porous media, etc): when zooming on a single perturbation pattern, one often ends with a periodic problem set on an infinite strip. We present a unified framework that enables a systematic treatment of such problems in the context of periodic strips. We provide existence and uniqueness of solutions in our weighted Sobolev spaces. This gives a refined description of solution’s behavior at infinity which is of importance in the multi-scale context. The isomorphisms are valid for any relative integer m, any p in \((1,\infty )\), and any real \(\alpha \) out of a countable set of critical values for the Stokes, the biharmonic and the Laplace operators.KeywordsPeriodic infinite stripWeighted Sobolev spacesHardy inequalityIsomorphismsLaplace operatorStokes equationsGreen functionBoundary layers