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Session 39. Contributed talks
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Extension theorems dealing with weighted Orlicz-Slobodetskii space |
Agnieszka Kałamajska, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
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The talk is based on joint works with Raj Narayan Dhara
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Having given weight \({\rho}=\tau \left(
\operatorname{dist}(x,\partial \Omega ) \right)\) defined on
Lipschitz boundary domain \(\Omega\) and Orlicz function \(R\), we
construct the weight \(\omega_{{{\rho}}}(\cdot,\cdot)\) defined on
\(\partial \Omega\times \partial \Omega\) and extension operator
\(Ext\) from certain subspace of weighted Orlicz--Slobodetski space
\(Y^{R,R}_{\omega_{{\rho}}}(\partial \Omega)\) subordinated to the
weight \(\omega_{{\rho}}\) to Orlicz--Sobolev space
\(W^{1,R}_{{\rho}}(\Omega)\). The weight
\(\omega_{{{\rho}}}(\cdot,\cdot)\) is independent of \(R\). This
gives the new tool to deal with boundary value problems like:
\begin{align*}\label{jedan}
\begin{Bmatrix}
-\operatorname{div}\left( {\rho}(x)B(\nabla u(x)) \right) =f & \operatorname{in}& \Omega\\
u=g&\operatorname{in}&\partial\Omega .
\end{Bmatrix}
\end{align*}
with inhomogeneous boundary data provided in the weighted Orlicz
setting. Result is new in the unweighted Orlicz setting for general
function \(R\) as well as in the weighted \(L^p\) setting.
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References- R. N. Dhara, A. Kalamajska, On one extension
theorem dealing with weighted Orlicz-Slobodetskii space. Analysis
on cube , preprint available at:
\url{http://www.mimuw.edu.pl/badania/preprinty/preprinty-imat/?LANG=en
- R. N. Dhara, A. Kalamajska, On one extension
theorem dealing with weighted Orlicz-Slobodetskii space. Analysis
on Lipschitz subgraph and Lipschitz domain , preprint available
at:
\url{http://www.mimuw.edu.pl/badania/preprinty/preprinty-imat/?LANG=en
- A. Kalamajska, M. Krbec, Traces of
Orlicz-Sobolev functions under general growth restrictions ,
Math. Nachr. 286, No. 7 (2013), 730-742.
- M.-TH. Lacroix, Espaces de traces des espaces
de Sobolev-Orlicz, J. Math. Pures Appl. 53 (9) (1974),
439-458.
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