content_image/320338.jpg content_image/320339.jpg content_image/320340.jpg
Then we choose s$_{2}$ = s$_{2}$ ( γ ) > 0 such that ∀ s > s$_{2}$ all the terms of X$_{2}$ and X$_{6}$ can be absorbed into ‖ P + s z ‖ 2 L $^{2}$(Ω) , ‖ P − s z ‖ 2 L 2 $_{(Ω)}$, C$_{5}$ ‖∇ $_{x}$z ‖ 2 L 2 $_{(Ω)}$, C$_{5}$ ‖∇ $_{y}$z ‖ 2 L $^{2}$(Ω) and C$_{5}$s 3 ‖ z ‖ 2 L 2 $_{(Ω)}$. Therefore ∫ Ω | P$_{s}$z + is∂$_{t}$ϕz | 2 dxdydt ≤ 2 ∫ Ω | P$_{s}$z | $^{2}$dxdydt + C$_{6}$s 2 ∫ Ω | z | $^{2}$dxdydt, taking s > 0 sufficiently large, we have 

<img src='content_image/320338.jpg'>

Since z = 0 on Γ$_{x}$ ∪ Γ$_{y}$ , ∇ $_{y}$z = 0 and ∇ $_{x}$z = ( ∂$_{ν}$z ) · ν on Γ$_{x}$ , all the integrations on Γ$_{y}$ vanish in (3.28), we have 

<img src='content_image/320339.jpg'>

We obtain 

<img src='content_image/320340.jpg'>

