Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 105 / 1 / d

The Fredholm alternative for an elliptic Dirichlet problem says that either the homogeneous adjoint problem has only the zero solution, in which case $Lu=f$ has a unique solution for every admissible $f$, or the homogeneous kernels are nontrivial and finite-dimensional. In the latter case,is solvable exactly whenand any two solutions differ by an element of $\ker L$. Moreover $\dim \ker L=\dim \ker L^{\ast }$.

For $L=d^2/d{x}^2+1$ with homogeneous Dirichlet data at $0$ and $2\pi$, integration by parts shows that $L^{\ast }=L$. Its homogeneous kernel isbecause $A\cos x+B\sin x$ vanishes at both endpoints exactly when $A=0$.

The forcing obeys the orthogonality conditionThe Fredholm alternative for an elliptic Dirichlet problem therefore says that solutions exist, though they are not unique. Indeed,satisfies $u^{\prime \prime }+u=\cos x$ and both boundary conditions for every constant $C$.