The square root problem for second-order, divergence form operators with mixed boundary conditions on L p

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• 作者：Pascal Auscher ; Nadine Badr ; Robert Haller-Dintelmann…
• 关键词：Primary ; 35J15 ; 42B20 ; 47B44 ; Secondary ; 26D15 ; 46B70 ; 35K20 ; Kato’s square root problem ; Elliptic operators with bounded measurable coefficients ; Interpolation in case of mixed boundary values ; Hardy’s inequality ; Calderon–Zygmund decomposition
• 刊名：Journal of Evolution Equations
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：15
• 期：1
• 页码：165-208
• 全文大小：505 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1424-3202

文摘

We show that, under general conditions, the operator \({(-\nabla . \mu \nabla + 1)^{1/2}}\) with mixed boundary conditions provides a topological isomorphism between \({W^{1,p}_D(\Omega) {\rm and} L^p(\Omega)}\) , for \(p \in ]1,2[ \) if one presupposes that this isomorphism holds true for p?=?2. The domain \({\Omega}\) is assumed to be bounded, and the Dirichlet part D of the boundary has to satisfy the well-known Ahlfors–David condition, whilst for the points from \({\overline {\partial \Omega \setminus D}}\) the existence of bi-Lipschitzian boundary charts is required.