Lp estimates for Dirichlet-to-Neumann operator and applications

• Main Authors: Toufic El Arwadi, Toni Sayah
• Format: Article
• Language: English
• Published: Texas State University 2015-10-01
• Series: Electronic Journal of Differential Equations
• Subjects: Dynamic boundary condition; Dirichlet-to-Neumann operator; Lp estimation; finite element method
• Online Access: http://ejde.math.txstate.edu/Volumes/2015/258/abstr.html

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.