minimising a functional

• Nov 26th 2008, 05:26 AM
mbbx5va2
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.(Nod)
• May 5th 2009, 06:29 PM
Rebesques
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.