1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

[PDE] 2D Vibrating Plate (Unique Solution)

  1. Oct 8, 2013 #1
    We have a region [itex]Ω[/itex] in [itex]ℝ^2[/itex] with a smooth boundary. There is a plate of shape [itex]Ω[/itex] and clamped edges which is approximated by the following equation:
    $$\frac{∂^2u}{∂t^2}=-Δ^2u$$
    $$u(x,t)=0\hspace{4ex} x\in ∂Ω$$
    $$Du(x,t)\cdot\hat{n}=0\hspace{4ex} x\in ∂Ω$$
    [itex]\hat{n}[/itex] is the outward pointing unit vector on the boundary of [itex]Ω[/itex]. Moreover, we specify the following initial conditions:
    $$u(x,0)=g(x)$$
    $$u_t(x,0)=h(x)$$
    Given all of this, we wish to show our problem has at most one solution.

    So the way I went about this was to let [itex]u[/itex] and [itex]\tilde{u}[/itex] solve the problem. We can consturct a solution [itex]w=u-\tilde{u}[/itex] that solves the PDE with homogeneous initial data. If [itex]w\equiv 0[/itex] on [itex]Ω[/itex], then our solution is unique.

    I am using the second edition of Lawrence Evans' Partial Differential Equations, and they use an energy method to prove uniqueness of a solution of the wave equation with given boundary/initial data. They define energy and its derivative with respect to time to be the following:
    $$E(t):=\frac{1}{2}\displaystyle\int_Ωw^2_t(x,t)+|Dw(x,t)|^2dx$$
    $$\frac{d}{dt}E(t)=\displaystyle\int_Ωw_tw_tt+Dw\cdot Dw_tdx$$
    I have difficulty following the next step:
    $$\frac{d}{dt}E(t)=\displaystyle\int_Ωw_t(w_{tt}-Δw)dx$$
    From there they go on to say that [itex]\frac{d}{dt}E(t)=0[/itex] and a chain of intuitive observations leads to the desired [itex]w\equiv 0[/itex]. I am confused by two things:

    (1)Why does [itex]Dw\cdot Dw_t=-w_tΔw[/itex]? Where does the negative come from?

    (2)If I were to replicate this for the higher order problem I posted, would I have to find more derivatives of energy to prove uniqueness?

    Thanks a bunch
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: [PDE] 2D Vibrating Plate (Unique Solution)
  1. Unique solution (Replies: 3)

  2. Unique Solution Proof (Replies: 1)

Loading...