The Neumann spectrum and the constant zero mode

Since $\mathrm{Re}a(u,u)={\int }_{\mathrm{\Omega }}\mathrm{Re}(a^{ij}{D}_{j}u\overline{D_{i}u})dx$ and the integrand is at least $\theta \mathrm{\mid }Du{\mathrm{\mid }}^2\ge 0$ almost everywhere by [F1], the integral is nonnegative. If $a(u,u)=0$, then $0\le \theta {\int }_{\mathrm{\Omega }}\mathrm{\mid }Du{\mathrm{\mid }}^2\le \mathrm{Re}a(u,u)=0$, so ${\int }_{\mathrm{\Omega }}\mathrm{\mid }Du{\mathrm{\mid }}^2=0$ and [F2] gives $\mathrm{\mid }Du\mathrm{\mid }=0$ almost everywhere; conversely $Du=0$ a.e. makes the integrand vanish a.e. and hence $a(u,u)=0$. By [F3], $Du=0$ a.e. holds exactly when $u$ is constant on each connected component of $\mathrm{\Omega }$.

If $\mathrm{\Omega }$ has finitely many connected components, write them as ${\mathrm{\Omega }}_1,\dots ,{\mathrm{\Omega }}_m$. The componentwise constants ${\sum }_{j=1}^m{c}_j{\mathbf{1}}_{{\mathrm{\Omega }}_j}$ with $c_j\in \mathbb{K}$ form a linear subspace of $H^1(\mathrm{\Omega })$ of dimension $m$, because the components are nonempty disjoint open sets of positive finite measure and each indicator has zero weak gradient: every compactly supported test function meets only finitely many components, and its integral derivative on each component is zero; by step 1.1 this subspace is exactly the zero set of the quadratic form $\mathrm{Re}a$, that is, the kernel of the symmetric form $a$ on $H^1(\mathrm{\Omega })$.

Weak Neumann eigenfunctions of eigenvalue $0$ are exactly the nonzero elements of that kernel: $a(u,v)=0$ for all $v\in H^1(\mathrm{\Omega })$ implies $a(u,u)=0$, hence $u$ is componentwise constant by step 1.1; conversely a componentwise constant $u$ has $Du=0$ a.e., so $a(u,v)={\int }_{\mathrm{\Omega }}{a}^{ij}{D}_{j}u\overline{D_{i}v}dx=0$ for every $v\in H^1(\mathrm{\Omega })$, and every nonzero constant function supplies such an eigenfunction. Thus $0$ is the lowest weak Neumann eigenvalue, since testing any weak eigenpair at its eigenfunction gives a nonnegative eigenvalue. Its eigenspace consists of the componentwise constants in $H^1(\mathrm{\Omega })$ and has dimension $m$ when there are exactly $m$ components.

The mean-zero refinement requires the extra hypotheses: if $\mathrm{\Omega }$ is connected and a Sobolev extension domain, [F4] supplies a positive first level on the global mean-zero subspace. Without connectedness this can fail: on a domain with several components, a nonzero componentwise constant such as ${\mathbf{1}}_{{\mathrm{\Omega }}_1}-\frac{\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }}{\mathrm{\mid }{\mathrm{\Omega }}_2\mathrm{\mid }}{\mathbf{1}}_{{\mathrm{\Omega }}_2}$ has global mean zero and zero form value, so no positive lower bound on the mean-zero space follows from the present hypotheses.