Hi, i'm having alot of trouble knowing what to do here. Not sure how i can 'minimize the functional' There is a slightly similar example in my notes entitled 'Model diffusion problem' where the differential equation is multiplied by a test function v and then integration by parts is used
let u belong to the space of twice differentiable functions on the open interval (0,1) such that: -u_xx=f (2nd derivative wrt x) where u(0)=u(1)=0 Prove that u minimizes the functional: F with domain the sobolev space H with subscript 0 and superscript 1 on the open interval omega=(0,1) and the range the real line. v belongs to domain
F(v)=1/2*int{(dv/dx)^2} - int{f(x)v(x)}dx each integral is from 0 to 1.
(Sorry i should learn to use latex.) Any suggestions as to which direction i should go with this would be much appreciated. thanks