A real variable method for the cauchy transform on graphs

1988 ◽  
pp. 31-70
Author(s):  
Takafumi Murai
Keyword(s):  
Real Variable ◽  
Variable Method ◽  
Cauchy Transform ◽  
Real Variable Method

scholarly journals $ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Bojing Shi
Keyword(s):  
Lipschitz Constant ◽  
Elliptic Problems ◽  
Elliptic Systems ◽  
Real Variable ◽  
Lipschitz Domains ◽  
Second Order Elliptic Problems ◽  
Bounded Lipschitz Domains ◽  
Drift Terms ◽  
Scalar Equations ◽  
Real Variable Method

<p style='text-indent:20px;'>In this paper, we establish the <inline-formula><tex-math id="M1">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates for solutions of second order elliptic problems with drift terms in bounded Lipschitz domains by using a real variable method. For scalar equations, we prove that the <inline-formula><tex-math id="M2">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates hold for <inline-formula><tex-math id="M3">\begin{document}$ \frac{3}{2}-\varepsilon&lt;p&lt;3+\varepsilon $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M4">\begin{document}$ d\geq3 $\end{document}</tex-math></inline-formula>, and the range for <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula> is sharp. For elliptic systems, we prove that the <inline-formula><tex-math id="M6">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates hold for <inline-formula><tex-math id="M7">\begin{document}$ \frac{2d}{d+1}-\varepsilon&lt;p&lt;\frac{2d}{d-1}+\varepsilon $\end{document}</tex-math></inline-formula> under the assumption that the Lipschitz constant of the domain is small.</p>

Adjoint variable method for two-dimensional plasmonic structures

2012 ◽  
Vol 37 (16) ◽  
pp. 3453 ◽  
Author(s):  
O. S. Ahmed ◽  
M. H. Bakr ◽  
X. Li ◽  
T. Nomura
Keyword(s):  
Variable Method ◽  
Two Dimensional ◽  
Adjoint Variable Method ◽  
Adjoint Variable ◽  
Plasmonic Structures
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