Electron. J. Diff. Equ., Vol. 2015 (2015), No. 258, pp. 1-8.

Lp estimates for Dirichlet-to-Neumann operator and applications

Toufic El Arwadi, Toni Sayah

Abstract:
In this article, we consider the time dependent linear elliptic problem with dynamic boundary condition. We recall the corresponding Dirichlet-to-Neumann operator on $\Gamma$ denoted by $-\Lambda_\gamma$. Then we show that when $\gamma=1$ near the boundary, $\Lambda_\gamma-\Lambda_1$ is bounded by $\gamma-1$ in $L^p(\Omega)$ norm. This result is a generalization of the bound with the $L^\infty(\Omega)$ norm and is applicable for comparing the Dirichlet to Neumann semigroup and the Lax semigroup. Finally, we present numerical experiments for validation of our results.

Submitted September 5, 2015. Published October 2, 2015.
Math Subject Classifications: 47D06, 47A99, 35J15, 35M13, 65M60.
Key Words: Dynamic boundary condition; Dirichlet-to-Neumann operator; Lp estimation; finite element method.

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Toufic El Arwadi
Department of Mathematics and computer science
Faculty of Science, Beirut Arab university
P.O. Box: 11-5020, Beirut, Lebanon
email: t.elarwadi@bau.edu.lb
Toni Sayah
Research unit "EGFEM", Faculty of sciences
Saint-Joseph University
B.P. 11-514 Riad El Solh
Beirut 1107 2050, Lebanon
email: toni.sayah@usj.edu.lb

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