I am working on problem 9.3.1 from Introduction to Differential Equations bij Peter J. Olver. I have just been introduced to self-adjointness and positive (semi) definiteness but I find it quite hard to understand. The problem is as follows:
Consider the boundary value problem $$-u''=x, u(0)=u(1)=0.$$
Now I need to
(a) find the solution
(b) write down a minimization principle that characterizes the solution
(c) find the value of the minized quadratic functional on the solution
(d) write down at least two other functions that satisfy the boundary conditions and check they produce larger values for the energy,
and honestly I have no clue where to even start.
(a) I suppose we can use the underlying differential operator $S= D^{*} \circ D = - D^{2}$ which is self-adjoint since we have Dirichlet boundary conditions. Now we can also use the fact that if $L$ is a linear operator between inner product spaces with adjoint $L^{*}$ and we assume the kernel just consists of the zero function, we then know $$Q[u] = \frac{1}{2} ||| L[u] |||^{2} - \langle f,u \rangle,$$ for $-u''=f(x)$, where the triple brackets indicate $\langle\langle \cdot, \cdot \rangle\rangle$ (i.e. the inner product of $L^{*}$) has a unique minimizer $u_{*}$. which is a solution to S[u]=f.
In our case this would then become
$$Q[u] = \frac{1}{2} ||| u' |||^{2} - \langle x,u \rangle = \int_{a}^{b} \left[ \frac{1}{2} u'(x)^{2} - xu(x) \right] dx.$$
Is this correct and if so: how to proceed?
Firstly let's solve the ODE. Integrating twice and inserting the boundary conditions we get that: $$u_0(x)=\frac16 x- \frac16 x^3 $$
Next we see that the ODE is a Euler-Lagrange equation associated with (for instance) the functional $Q[u] = \int_0^1 \left(\frac12 u'(x)^2 -xu(x) \right)dx$.
Inserting our solution we get that $Q[u_0]=-\frac{1}{90}$.
Lastly we need to calculate $Q$ for two functions that satisfy the boundary conditions. Let's take the functions $u_1(x)=x(1-x)$ and $u_2(x)=\sin(\pi x)$. Inserting into $Q$ we get: $$Q[u_1]=\frac{1}{12} $$ $$Q[u_2] = \frac{\pi^3-4}{4\pi}\approx 2.15 $$ Finally we note that $Q[u_0]<Q[u_1]<Q[u_2]$ as expected.