Math 6420 • PDE

Consider the fourth-order differential equation
\[
\begin{aligned}
u^{(4)} &= f, && \text{on } (0,1),\\
u(0) &= u(1), && u'(0)=u'(1),\\
u''(1) &= -2, && u^{(3)}(1)=1.
\end{aligned}
\]
\begin{enumerate}
\item[(a)] Find the least restrictive constraint(s) on \(f\in L^2(0,1)\) under which a solution
to the problem exists.
\item[(b)] With fixed \(f\) chosen such that a solution exists, is the solution unique? If the
solution is unique, prove it. Otherwise provide a counterexample.
\end{enumerate}