= Solution
\b[Green identity.] With $n$ the outward unit normal, the <product rule> gives $\nabla\cdot(u\nabla v)=u\nabla^2v+\nabla u\cdot\nabla v$. Integrating over $V$ and applying the <divergence theorem> yields <Green's first identity>:
$$
\boxed{\int_V(u\nabla^2v+\nabla u\cdot\nabla v)\,dV=\int_Su\partial_nv\,dS.}
$$
\b[Uniqueness for nonzero mass.] Suppose two real solutions have the same prescribed boundary data, and let $w$ be their difference. The spatial equation called the <Klein-Gordon equation> in the question is the <modified Helmholtz equation>, so $\nabla^2w=m^2w$. With $u=v=w$, <Green's first identity> becomes
$$
\int_V(|\nabla w|^2+m^2w^2)\,dV=\int_Sw\partial_nw\,dS.
$$
For identical <Dirichlet boundary conditions>, $w=0$ on $S$; for identical <Neumann boundary conditions>, $\partial_nw=0$ there. Either makes the right-hand side zero. When $m$ is real and nonzero, both integrands on the left are nonnegative, forcing $w=0$. Thus
$$
\boxed{\text{For }m\ne0,\text{ either type of boundary data determines at most one solution.}}
$$
This is the required uniqueness assertion; existence is understood for admissible smooth data on a suitable bounded region. Complex-valued solutions obey the same result by treating their real and imaginary parts separately.
\b[Zero mass.] For $m=0$, the argument gives $\nabla w=0$, hence $w=C$ on the connected region. <Dirichlet boundary conditions> still force $C=0$, but <Neumann boundary conditions> leave an arbitrary additive constant:
$$
\boxed{m=0:\quad\text{Dirichlet uniqueness; Neumann uniqueness only up to a constant.}}
$$
The prescribed normal data in the zero-mass case must also satisfy $\int_S\partial_n\Phi\,dS=0$, by the <divergence theorem>. For nonzero $m$, that flux can instead equal $m^2\int_V\Phi\,dV$.
\b[Energy minimization.] Let $\eta=w-v$ for the two functions in the last part. They share their boundary values, so $\eta=0$ on $S$. Since $v$ is a <harmonic function>, <Green's first identity> gives
$$
\int_V\nabla v\cdot\nabla\eta\,dV=\int_S\eta\partial_nv\,dS-\int_V\eta\nabla^2v\,dV=0.
$$
Expand the square in the <Dirichlet energy> of $w=v+\eta$:
$$
\int_V|\nabla w|^2\,dV=\int_V|\nabla v|^2\,dV+2\int_V\nabla v\cdot\nabla\eta\,dV+\int_V|\nabla\eta|^2\,dV.
$$
The cross term vanishes, so
$$
\boxed{\int_V|\nabla w|^2\,dV-\int_V|\nabla v|^2\,dV=\int_V|\nabla(w-v)|^2\,dV\geq0.}
$$
Equality requires $w-v$ to be constant, and the shared boundary values then make it zero. This is the <Dirichlet principle>: among functions with fixed boundary values, the <harmonic function> minimizes <Dirichlet energy>.
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