Lp estimates for Dirichlet-to-Neumann operator and applications
Toufic El Arwadi, Toni Sayah · DOAJ (DOAJ: Directory of Open Access Journals) · 2015
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.