Maximizing a functional with respect to a function

Question

I want to find the function $u(r)$ maximizing the functional $F(u(r))$:

$$F(u(r))=\int_{-R}^{R} u(r)^2 dr$$

subject to the constraints below:

$$\int_{-R}^{R}u(r) dr= U_0 $$ $$u(-R)=u(R)=0$$ in which $U_o$ is a constant.

Update: $u(r)$ is twice differentiable.

Answer 1

The supremum of the functional $F(u)$ is $+\infty$. Consider the sequence of functions defined by $$ u_n(x)=U_0 n/R\quad \textrm{if } -R/(2n)\le x\le R/(2n),\quad u_n(x)=0 \textrm{ elsewhere} $$ The functions satisfy the constraints, but $$ \int_{-R}^R u_n(x)^2 dx=nU_0^2/R $$ tending to infinity. The answer dos not change if we add the hypothesis that $u$ is differentiable (even infinite times). One can just substitute $u_n$ with a smoothed version of the "peak function"