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A disconnected Neumann domain has a multiple zero eigenvalue

Example

Assume the Axiom of Choice (AC) and Countable Choice (CC). The explicit CC premise is used by Uniformly elliptic divergence-form operators and their sesquilinear forms, while AC matches the cited componentwise-constancy and mean-zero Neumann results as currently stated. Let $Ω \subset R^{2}$ be the union of two disjoint open discs $B_{1}, B_{2}$ (a nonempty bounded open set with exactly two connected components), and let $a (u, v) = \int_{Ω} D u \cdot \overline{D v} d x$ on $H^{1} (Ω)$ (Uniformly elliptic divergence-form operators and their sesquilinear forms with $a^{i j} = δ^{i j}$ ). Then $a (u, u) \geq 0$ with equality if and only if $u$ is constant on each component, so the weak Neumann eigenvalue $0$ has the two-dimensional eigenspace ${c_{1} 1_{B_{1}} + c_{2} 1_{B_{2}} : c_{1}, c_{2} \in K} .$ The multiplicity of the zero eigenvalue equals the number of connected components. Consequently Poincare-Wirtinger with the global mean fails on $Ω$ : the mean-zero function $∣ B_{2} ∣ 1_{B_{1}} - ∣ B_{1} ∣ 1_{B_{2}}$ has zero energy, so on the mean-zero subspace the Rayleigh infimum is $0$ , not positive, and the connectedness hypothesis of The first positive Neumann eigenvalue has the mean-zero Rayleigh characterisation cannot be dropped.

Facts & Assumptions

Given: the Axiom of Choice and Countable Choice; two nonempty disjoint open discs $B_{1}, B_{2} \subset R^{2}$ of finite positive area $∣ B_{1} ∣, ∣ B_{2} ∣$ with $Ω = B_{1} \cup B_{2}$ ; the Neumann form $a (u, v) = \int_{Ω} D u \cdot \overline{D v} d x$ on $H^{1} (Ω)$ .

[F1]

The zero mode of the principal Neumann form: for a bounded open set the form $a$ is nonnegative, and $a (u, u) = 0$ if and only if $D u = 0$ a.e., if and only if $u$ is constant on every connected component (The Neumann spectrum and the constant zero mode, A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere, Zero weak gradient gives componentwise constants).

[F2]

Connected components: each disc $B_{j}$ is connected, the two discs are disjoint open sets, so $Ω$ has exactly the two connected components $B_{1}, B_{2}$ ; the componentwise constants $c_{1} 1_{B_{1}} + c_{2} 1_{B_{2}}$ (equal on $Ω$ to $c_{j}$ on $B_{j}$ ) form a two-dimensional subspace of $H^{1} (Ω)$ (Connected components, quasicomponents, and totally disconnected spaces, Integer-order Sobolev spaces and their norms, Uniformly elliptic divergence-form operators and their sesquilinear forms).

[F3]

Integrals of indicators: $\int_{Ω} 1_{B_{j}} d x = ∣ B_{j} ∣$ and integrals are additive, computed in the almost-everywhere class convention (Integral over a measurable subset, The space $L^{p} (μ)$ as the quotient by null functions).

[F4]

The positive first level on the global mean-zero space requires connectedness and the extension-domain property; on a disconnected domain the mean-zero subspace is larger and the positivity is not forced (The first positive Neumann eigenvalue has the mean-zero Rayleigh characterisation).

Verification

technique · direct

1.1F1F2givenalgebra

Specialising [F1] to $Ω = B_{1} \cup B_{2}$ gives $a (u, u) = ∥ D u ∥_{L^{2}}^{2} \geq 0$ , with equality exactly when $D u = 0$ a.e., i.e. exactly when $u$ is constant on each of the two components. Hence the kernel of the form on $H^{1} (Ω)$ , that is the zero eigenspace of the weak Neumann problem, is the space ${c_{1} 1_{B_{1}} + c_{2} 1_{B_{2}}}$ of [F2].

2.1F2step 1.1algebra

The two functions $1_{B_{1}}$ and $1_{B_{2}}$ are nonzero linearly independent classes and lie in the kernel by step 1.1, so the zero eigenspace is exactly two-dimensional; the multiplicity of the eigenvalue $0$ equals the number $2$ of connected components of $Ω$ .

2.2F3step 1.1givenalgebra

Put $g : = ∣ B_{2} ∣ 1_{B_{1}} - ∣ B_{1} ∣ 1_{B_{2}}$ . By [F2] it is componentwise constant, hence a nonzero element of $H^{1} (Ω)$ with zero weak gradient, and step 1.1 gives $a (g, g) = 0$ . Its global mean is $∣ B_{2} ∣ \int_{Ω} 1_{B_{1}} - ∣ B_{1} ∣ \int_{Ω} 1_{B_{2}} = ∣ B_{2} ∣ ∣ B_{1} ∣ - ∣ B_{1} ∣ ∣ B_{2} ∣ = 0$ by [F3], so $g$ lies in the mean-zero subspace $V$ and is nonzero with vanishing Rayleigh quotient. Therefore $\inf_{u \in V ∖ {0}} a (u, u) / ∥ u ∥_{L^{2}}^{2} = 0$ on this $Ω$ .

3.1F1F4step 2.2given∎

Consequently no Poincare-Wirtinger inequality with the global mean and a positive constant can hold on the disconnected set $Ω$ : such an inequality would bound $∥ g ∥_{L^{2}}$ by a positive multiple of $∥ D g ∥_{L^{2}} = 0$ for the nonzero function $g$ of step 2.2. This shows that the connectedness hypothesis in [F4] cannot be dropped, while the two-dimensional zero eigenspace of step 2.1 shows that the multiplicity of the Neumann eigenvalue $0$ equals the number of connected components.

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Sources

Richard S. Laugesen, Spectral Theory of Partial Differential Equations (University of Illinois lecture notes, arXiv:1203.2344, complete 120 pages) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page notes) (standard reference, not scraped)
Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, 2020, complete 158-page graduate notes) (standard reference, not scraped)