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Math Help - minimising a functional

  1. #1
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    minimising a functional

    Hi, if u belongs to C2(0,1)=space of twice differentiable continuous functions on (0,1). Also u(1)=u(0)=0 are boundary conditions with u(x)=-u double prime =f (ie 2nd derivative of u wrt x). Prove that u minimizes the functional F where domain is sobolev space with m=1 on (0,1) (subscript 0) and range is real line.

    F(v)=1/2*int(v prime)^2 - int(f(x)v(x))dx (Integrating over 0 and 1)

    From above we know u is solution of strong problem which implies it is the solution of the variational problem.

    Now i've managed to represent F(v) as 1/2a(v,v)-a(u,v) now to show this is minimized i was thinking i could bound it above by something with a constant in and then choose a large constant. Looking at coercivity and continuity these could be applied but still not sure which way of going about it. Any suggestions would be much appreciated. Thanks.
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  2. #2
    Super Member Rebesques's Avatar
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    You mean F(v)=a(u',v')/2-a(f,v), where a(v,w)=\int vw is the dot product on W^{1,2}_0=H^1_0.


    You need to prove the following:



    1. A minimizer for F is a solution of the DE.
    2. A solution of the DE is a minimizer for F.




    For 1, notice that the DE is the Euler-Lagrange equation of F.
    For 2, calculate F(u+tv)-F(u) and use the DE to show this is equal to  \int t^2v'^2/2. So F attains a minimum on u.
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