Minimum Problem N.2

[Camillo]

Let be $F(v) = int_0^1[hat v^2(t)+v^2(t)+v(t) hat v(t)] dt $.
Show, explaining the reason , if $F(v)$ has minimum in the following sets and, in positive case, calculate it :
$A=(v in C^1 [0,1] : v(0)=0, v(1)=1 )$.
$B =(v in C^1[0,1] : v(1)=1 )$.

P.S. I had problems in writing the Functional $F(v) $ , note that $ hat v(t) $ means the first derivative of $v(t)$, that is $ v'(t) $.

[Luca.Lussardi]

I try for the first case. In the case $A$ we easily get, integrating by parts, $\int_0^1v v'=1-\int_0^1v v'$ and then $\int_0^1v v'=1/2$. Thus $F(u)=1/2+||v||^2$, where $||\cdot||$ is the norm in $H^1(0,1)$. Now since $0$ does not belong to $A$, if $A$ is closed in $H^1(0,1)$ (is it true?) then, since it is convex, $G(v)=||v||^2=||v-0||^2$ has a unique minimum on $A$ by projection theorem in Hilbert spaces.

[Camillo]

I know that generally $hat v $ represents the Fourier transform of $ v $ but MathMl does not accept the sign " ' " with exponent

[gugo82]

"Camillo":P.S. I had problems in writing the Functional $F(v) $ , note that $ hat v(t) $ means the first derivative of $v(t)$, that is $ v'(t) $.

At first sight I thought $\hat{v}$ to be the Fourier transform of $v$...

[gac1]

I do not see any simple solution.
My outline of a standard solution is the following.

1) The Lagrangian $L(v, z) = z^2+v^2+vz$ is superlinear and convex w.r.t. $z$, so the existence of minimizers follows from the direct method of the calculus of variations. These minimizers satisfy the Euler-Lagrange condition.

2) Euler-Lagrange condition gives $v''-v=0$; together with the final condition $v(1)=1$, we get the functions
$v_c(t) = c e^t + e(1-ce)e^{-t}$.

3) In the set $A$, the initial condition $v(0)=0$ selects only one function in the class $\{v_c\}$, i.e.
$v(t) = \frac{e}{e^2-1} (e^t - e^{-t})$.

4) In the set $B$ we can minimize $F[v_c]$ w.r.t. the parameter $c$.
A simple computation gives $F[v_c] = \frac{1}{2}(e^2-1)[c^2+(1-ce)^2]$, which is minimized by $c=\frac{e}{e^2+1}$.