 Solution

The Generalized Stokes theorem applied to the closed differential one-form givesEquivalently, the global relation isThe identical outward Neumann boundary data make the three known transforms equal:To reduce the unknown traces to a single function, use rotation invariance and uniqueness. For $\lambda >0$ the difference $u$ of two solutions has zero outward derivative andby Green's first identity. Thus $u=0$. Rotation by $a$ preserves the equation, the domain and the oriented boundary data, so it preserves $q$; hence $q^{(1)}=q^{(2)}=q^{(3)}=q$ and ${\Phi }_j=\Phi$. The global relation becomes the single equation with three spectral unknownsThe three unknowns are evaluations of the same boundary transform, not three freely specified boundary data.

The printed allowance of all real $\lambda$ requires a qualification. At $\lambda =0$, existence requires $3{\int }_{-l/2}^{l/2}f(s)ds=0$ by the divergence theorem, and solutions differ by a constant. Fixing a rotation-invariant normalization gives uniqueness and the same rotational conclusion. In fact the unnormalized solution is already invariant: its rotation difference is constant, and three successive rotations make three times that constant zero. For $\lambda <0$, uniqueness fails precisely when $-4\lambda$ is an eigenvalue of the Neumann Laplacian. At these parameters identical data do not force an arbitrary solution to be rotationally invariant. Averaging any solution over the three rotations produces a solution with the common trace used here; the difference is a homogeneous Neumann eigenfunction, or a sum of them. Part (e) describes both solvability and this remaining freedom rather than assuming uniqueness for every real parameter.