Lower bounds for the first eigenvalue of the Laplacian with zero magnetic field in planar domains

Let $\Omega$ be a bounded, open, connected domain with smooth boundary $\partial\Omega$ in a Riemannian manifold $(M,g)$ and let $A$ be a smooth real one-form on $\Omega$, (the potential one-form). Define a connection $\nabla^{A}$ on the space of complex-valued functions $C^{\infty}(\Omega,{\bf C})$ as follows:

$$\nabla^{A}_{X}u=\nabla_{X}u-iA(X)u,$$

for all vector fields $X$ on $\Omega$, where $\nabla$ is the Levi-Civita connection of $M$. The magnetic Laplacian with potential $A$ is the operator acting on $C^{\infty}(\Omega,{\bf C})$:

$$\Delta_{A}=(\nabla^{A})^{\star}\nabla^{A}.$$

In ${\bf R}^{n}$ this gives explicitly, in the usual notation:

$$\Delta_{A}=(i\nabla+A^{\sharp})^{2},$$

where $A^{\sharp}$ is the dual vector field of $A$, the vector potential. The two-form $B=dA$ is the magnetic field; dually, in dimension $2$, $B$ is the vector field $B={\rm curl}A^{\sharp}$.

Scope of this paper is to discuss the spectrum of $\Delta_{A}$ for planar domains. Hence in what follows we take $\Omega\subset{\bf R}^{2}$.

The spectrum of the magnetic Laplacian has been studied extensively for Dirichlet boundary conditions ($u=0$ on $\partial\Omega$), and we denote by $\lambda_{1}^{D}(\Omega,A)$ the first eigenvalue. First we remark that, thanks to the diamagnetic inequality, one always has:

$$\lambda_{1}^{D}(\Omega,A)\geq\lambda_{1}^{D}(\Omega,0),$$

and in particular $\lambda_{1}^{D}(\Omega,A)>0$. For planar domains and constant magnetic field (that is, $dA=B$ and $\lvert{B}\rvert$ constant), a Faber-Krahn inequality holds, in the sense that the first eigenvalue of a planar domain is minimized by that of the disk of the same area (see [5]). Estimates for sums of eigenvalues can be found in [9].

However in this paper we deal with magnetic Neumann boundary conditions, that is we impose $\nabla^{A}_{N}u=0,$ on the boundary, where $N$ is the inner unit normal to $\partial\Omega$. It is known that then $\Delta_{A}$ admits a discrete spectrum

$$0\leq\lambda_{1}^{N}(\Omega,A)\leq\lambda_{2}^{N}(\Omega,A)\leq\dots$$

diverging to $+\infty$. The first eigenvalue has the following variational characterization:

$$\lambda_{1}^{N}(\Omega,A)=\min_{u\in H_{1}(\Omega)\setminus\{0\}}\frac{\int_{\Omega}\lvert{\nabla^{A}u}\rvert^{2}dx}{\int_{\Omega}|u|^{2}dx}.$$

(1)

For computing lower bounds the diamagnetic inequality is of no use; in fact it gives:

$$\lambda_{1}^{N}(\Omega,A)\geq\lambda_{1}^{N}(\Omega,0)=0,$$

because $\lambda_{1}^{N}(\Omega,0)$ is simply the first eigenvalue of the usual Neumann Laplacian, which is zero (the associated eigenspace being spanned by the constant functions). There are fewer estimates in this regard; let us first discuss the case of a constant magnetic field $\lvert{B}\rvert=B_{0}>0$ on planar domains. The paper [4] gives a lower bound of $\lambda_{1}^{N}(\Omega,A)$ in terms of the inradius of $\Omega$, $\lambda_{1}^{N}(\Omega,0)$ and of course $B_{0}$. Asymptotic expansions as $\lvert{B}\rvert\to\infty$ are obtained in [7]. We also mention the paper [6] which investigates the validity of a reverse Faber-Krahn inequality for constant magnetic field $B_{0}$, that is: is it true that $\lambda_{1}^{N}(\Omega,A)$ is always bounded above by that of a disk with equal volume ? It is proved there that this inequality is true when $B_{0}$ is either sufficiently small or sufficiently large, but the general case is still open in the simply connected case.

$\bullet\quad$In this paper we prove three lower bounds for the first eigenvalue of planar domains under Neumann conditions, when the magnetic field is identically zero. Since this will be the only boundary condition we consider, from now on we will simply write $\lambda_{1}(\Omega,A)$ instead of $\lambda_{1}^{N}(\Omega,A)$.

Let us first clarify the circumstances under which the first eigenvalue might be positive even if the magnetic potential is a closed one-form on $\Omega$. This is intimately related to a phenomenon in quantum mechanics predicted in 1959 and known as Aharonov-Bohm effect, which has also experimental evidence: a particle travelling a region in the plane might be affected by the magnetic field even if this is identically zero on its path. In fact what the particle ”feels” is not the magnetic field but, rather, the magnetic potential $A$, provided that $A$ is closed but not exact, and that the flux of $A$ around the pole may assume non-integer values (see below for the precise condition).

Let us be more precise. From the definition we see that, if $A=0$, the spectrum of $\Delta_{A}$ coincides with the spectrum of the usual Laplacian under Neumann boundary conditions. The same is true when $A=df$ is an exact one-form, by the well-known gauge invariance of the magnetic Laplacian. This fundamental property states that the spectrum of $\Delta_{A+df}$ is the same as the spectrum of $\Delta_{A}$, for any $f\in C^{\infty}(\Omega)$, which follows from the identity:

$$\Delta_{A}e^{-if}=e^{-if}\Delta_{A+df}$$

showing that $\Delta_{A}$ and $\Delta_{A+df}$ are unitarily equivalent.

On the other hand, if the magnetic field $B=dA$ is non-zero, then $\lambda_{1}(\Omega,A)$ is strictly positive. One could then ask if $\lambda_{1}(\Omega,A)$ has to vanish whenever the magnetic field is zero, that is, whenever $A$ is a closed one-form.

To that end, let $c$ be a closed curve in $\Omega$ (a loop). The quantity:

$$\Phi^{A}_{c}=\dfrac{1}{2\pi}\oint_{c}A$$

is called the flux of $A$ across $c$ (we assume that $c$ is travelled once, and we will not specify the orientation of the loop; this will not affect any of the statements, definitions or results which we will prove in this paper).

It turns out that

$\bullet\quad$$\lambda_{1}(\Omega,A)=0$ if and only if $A$ is closed and the cohomology class of $A$ is an integer, that is, the flux of $A$ around any loop is an integer.

This was first observed by Shigekawa [12] for closed manifolds, and then proved in [8] for manifolds with boundary. This remarkable feature of the magnetic Laplacian shows its deep relation with the topology of the underlying manifold $\Omega$. In this paper we will focus precisely on the situation where the potential one form is closed, and we will then give two lower bounds for the first eigenvalue $\lambda_{1}(\Omega,A)$.

Let us then recall a few previous results when the magnetic field is assumed to vanish. A lower bound for a general Riemannian cylinder (i.e. the surface ${\bf S}^{1}\times(0,L)$ endowed with a Riemannian metric) and zero magnetic field has been given in [3], and is somewhat the inspiration of this work: one of two main results here is in fact to improve such bound when $\Omega$ is a doubly connected planar domain.

Directly related to the Aharonov-Bohm effect, we mention the papers [1] and [10] which investigate the behavior of the spectrum of a domain with a pole $\Omega\setminus\{a\}$ when the pole $a$ approaches the boundary, for Dirichlet boundary conditions. We remark here that the pole is a distinguished point $a=(a_{1},a_{2})$ and the potential is the harmonic one-form:

$$A_{a}=\frac{1}{2}\Big{(}-\dfrac{x_{2}-a_{2}}{(x_{1}-a_{1})^{2}+(x_{2}-a_{2})^{2}}dx_{1}+\dfrac{x_{1}-a_{1}}{(x_{1}-a_{1})^{2}+(x_{2}-a_{2})^{2}}dx_{2}\Big{)}$$

which has flux $\frac{1}{2}$ across any closed curve enclosing $a$, giving rise to a magnetic field which is a Dirac distribution concentrated at the pole $a$ (therefore, the magnetic field indeed vanishes on $\Omega\setminus\{a\}$). The magnetic Laplacian $\Delta_{A_{a}}$ acting on $\Omega\setminus\{a\}$ is often called an Aharonov-Bohm operator. One could think to a domain with a pole as a doubly connected domain for which the inner boundary curve shrinks to a point.

We will in fact give a lower bound for the first eigenvalue of Aharonov-Bohm operators with many poles, and Neumann boundary conditions (see Theorem 3).

The Aharonov-Bohm operators play an interesting role in the study of minimal partitions, see chapter 8 of [2].

For Neumann boundary conditions, we mention the paper [8], where the authors study the multiplicity and the nodal sets corresponding to the ground state for non-simply connected planar domains with harmonic potential. For doubly connected domains, it is shown that $\lambda_{1}(\Omega,A)$ is maximal precisely when $\Phi^{A}$ is congruent to $\frac{1}{2}$ modulo integers (this fact is no longer true when there are more than two holes). The proof relies on a delicate argument involving the nodal line of a first eigenfunction and the conclusion does not follow from a specific comparison argument, or from an explicit lower bound.

The focus of this paper is on lower bounds for multiply connected planar domains and zero magnetic field defined by the closed potential form $A$. By what we have just said, it is clear that estimating the first eigenvalue is a trivial problem if $\Omega$ is simply connected, because then any closed one-form is automatically exact, and therefore $\lambda_{1}(\Omega,A)=0$ by gauge invariance. Therefore, we restrict our study to domains with $n$ holes, with $n\geq 1$.

In this paper we will prove: an improved lower bound for doubly connected domains; a general lower bound for multiply connected domains with an arbitrary number of convex holes; a lower bound for a general convex domain with an arbitrary number of punctures. Let us describe these results in detail.

Let us start from doubly connected domains ($n=1$) hence domains of type:

$$\Omega=F\setminus\bar{G},$$

where $F$ and $G$ are open and smooth. We assume $F$ and $G$ convex. Let $\Phi^{A}$ be the flux of the closed potential $A$ around the inner boundary curve $\partial G$: by Shigekawa result, the lower bound is simply zero when $\Phi^{A}$ is an integer. Then, to hope for a positive lower bound, we need to measure how much $\Phi^{A}$ is far from being an integer, and the natural invariant will then be:

$$d(\Phi^{A},{\bf Z})=\min\{\lvert{\Phi^{A}-k}\rvert:k\in{\bf Z}\}.$$

The second important ingredient for our lower bounds is the ratio $\frac{\beta}{B}$ between the minimal width and the maximal width of $\Omega$. To be more precise, let us say that the line segment $\sigma\subset\Omega$ is an orthogonal ray if it hits the inner boundary $\partial G$ orthogonally. By definition, the minimal width $\beta$ (resp. maximal width $B$) of $\Omega$ is the minimal (resp. maximal ) length of an orthogonal ray contained in $\Omega$:

Note that the ratio $\frac{\beta}{B}$ is invariant under homotheties, and reaches its largest value $1$ whenever the boundary components are parallel curves.

In Theorem 2 of [3] we prove the lower bound:

$$\lambda_{1}(\Omega,A)\geq\dfrac{4\pi^{2}}{\lvert{\partial F}\rvert^{2}}\dfrac{\beta(\Omega)^{2}}{B(\Omega)^{2}}d(\Phi^{A},{\bf Z})^{2}.$$

(2)

We insist on the fact that if $\frac{\beta}{B}$ is bounded below away from zero we get a positive, uniform lower bound even if $\beta$ tends to zero. Think for example to a concentric annulus $\Omega$ of radii $1$ and $1+\beta$; then $\frac{\beta}{B}=1$ and as $\beta\to 0$ the lower bound will approach $4\pi^{2}d(\Phi^{A},{\bf Z})^{2}$, a positive number, which is just the first eigenvalue of the unit circle.

This means that (for fixed perimeter) in order to get $\lambda_{1}$ small, the ratio $\frac{\beta}{B}$ (and not just $\beta$) has to be small.

Sharpness in terms of $\frac{\beta}{B}$. In [3] we showed that if $\frac{\beta}{B}$ is small then the first eigenvalue could indeed be small. We then looked for an example which could show that the dependance on $\frac{\beta^{2}}{B^{2}}$ is sharp, and we could not find it. Rather, in Examples 14 and 15 in [3], we constructed examples of domains such that $B$ is bounded below, say by $1$, $\lvert{\partial F}\rvert$ is bounded above, $\beta$ goes to zero and $\lambda_{1}(\Omega,A)$ goes to zero proportionally to $\beta$, for any non-integral flux. Therefore, if one could replace $\frac{\beta^{2}}{B^{2}}$ by the linear factor $\frac{\beta}{B}$ in (2), one would obtain a sharp inequality (with respect to $\frac{\beta}{B}$). See Figure 2 below for the example which shows sharpness.

This is in fact possible, and the theorem which follows should be regarded as the first main theorem of this paper.

Note that, modulo a factor of $4$, b) is formally identical to (2) with $\beta/B$ replacing $\beta^{2}/B^{2}$.

We observe that there is no positive constant $c$ such that

$$\frac{\beta(\Omega)}{B(\Omega)}\geq c\dfrac{\lvert{F}\rvert^{2}}{D(F)^{4}}$$

for all doubly convex annuli in the plane (otherwise, the lower bound would be independent on the inner hole, and this is impossible). This means that Theorem 1 is not a trivial consequence of (2).

In fact, the proof of Theorem 1 uses a suitable partition of $\Omega$ into overlapping annuli for which $\frac{\beta}{B}$ is, so to speak, as small as possible (see Section 2 below, and in particular Figure 3 for an example). Recall the $\delta$-interior ball condition:

given $x\in\partial F$, there is a ball of radius $\delta$ tangent to $\partial F$ at $x$ and entirely contained in $F$.

Here and for further applications, we say that the injectivity radius of $\partial F$ is ${\rm Inj}(\partial F)$ if $F$ satisfies the $\delta$-interior ball condition for any $\delta\leq{\rm Inj}(\partial F)$. If $\partial F$ is smooth, its injectivity radius is positive.

Finally we picture below the family of domains $\Omega_{\epsilon}$ which realize sharpness. $\Omega_{\epsilon}$ is the difference between two rectangles with parallel sides, with boundaries being $\epsilon$ units apart. Hence $\beta(\Omega_{\epsilon})=\epsilon$ and $B(\Omega_{\epsilon})$ is uniformly bounded above by $\sqrt{5}$.

We show in Section 2.8 that

$$\frac{\pi^{2}}{360\sqrt{5}}d(\Phi^{A},{\bf Z})^{2}\leq\dfrac{\lambda_{1}(\Omega_{\epsilon},A)}{\epsilon}\leq\frac{1}{10},$$

so that $\lambda_{1}(\Omega,A)$ goes to zero proportionally to $\epsilon\sim\frac{\beta}{B}$.

Now let $\Omega$ be an $n$-holed planar domain, which we write as follows:

$$\Omega=F\setminus(\bar{G}_{1}\cup\dots\cup\bar{G}_{n})$$

(4)

where the inner holes $G_{1},\dots,G_{n}$ are smooth, open and disjoint. We furthermore assume that $F,G_{1},\dots,G_{n}$ are convex. Note that:

$$\partial\Omega=\partial F\cup\partial G_{1}\cup\dots\cup\partial G_{n}.$$

We will call $\partial G_{1}\cup\dots\cup\partial G_{n}$ the inner boundary of $\Omega$. The minimal and maximal widths of $\Omega$ are defined as in the case $n=1$, namely $\beta$ is the minimal length of a line segment contained in $\Omega$ and hitting the inner boundary orthogonally, and the maximal length of such line segments is by definition the maximal width $B$.

It is clear that we could replace $B(\Omega)$ by the diameter of $F$, and $\beta(\Omega)$ by the invariant:

$$\tilde{\beta}(\Omega)=\min\{d(\partial G_{j},\partial G_{k}),d(\partial G_{h},\partial F):j\neq k,h=1,\dots,n\}.$$

In this section we give a lower bound of $\lambda_{1}(\Omega,A)$ when $\Omega$ has an arbitrary number of convex holes.

The proof uses a suitable decomposition of $\Omega$ into a finite union of annuli, and a lower bound proved in [3] for annuli whose outer boundary is star-shaped with respect to the inner boundary curve. A stronger estimate is proved when the inner holes are disks of the same radius (see Theorem 13).

The power $\frac{\beta^{4}}{B^{4}}$ in the previous estimate is probably not sharp; it appears to be there for technical reasons. By shrinking the inner boundary curves to points we obtain an estimate in terms of $\frac{\beta^{2}}{B^{2}}$, which has an interesting interpretation in terms of Aharonov-Bohm operators with many poles.

Precisely, we fix a convex domain $\Omega$ and choose $n$ points inside it, say $\mathcal{P}=\{p_{1},\dots,p_{n}\}$. Consider the punctured domain $\Omega\setminus\mathcal{P}$. Given a closed one-form $A$, we define:

$$\lambda_{1}(\Omega\setminus\mathcal{P},A)=\liminf_{\delta\to 0}\lambda_{1}(\Omega\setminus\mathcal{P}(\delta),A)$$

where $\mathcal{P}(\delta)$ is the $\delta$-neighborhood of $\mathcal{P}$ (it obviously consists of a finite set of disks of radius $\delta$). It is not our scope in this paper to investigate the convergence in terms of $\delta$; however, what we are looking at could be interpreted as the first eigenvalue of a Aharonov-Bohm operator with poles $p_{1},\dots,p_{n}$ and Neumann boundary conditions. The proof of the theorem in the previous section simplifies, to give a general lower bound in terms of the distance between the poles, and the distance of each pole to the boundary. To that end, define:

$$\left\{\begin{aligned} &\beta(\mathcal{P})=\min\{d(p_{j},p_{k}),d(p_{m},\partial\Omega):p_{j}\neq p_{k},p_{m}\in\mathcal{P}\}\\ &B(\mathcal{P})=\max\{d(p_{j},p_{k}),d(p_{m},\partial\Omega):p_{j}\neq p_{k},p_{m}\in\mathcal{P}\}\end{aligned}\right.$$

Of course $B({\mathcal{P}})$ could be conveniently bounded above by the diameter of $\Omega$. Let $A$ be as usual a closed one-form having flux $\Phi_{j}$ around the pole $p_{j}$. Then we have:

$\bullet\quad$In a forthcoming paper, we will give upper bounds for the Laplacian with zero magnetic field on multiply connected planar domains, which are closely related to the topology (number of holes) of the domain.

The rest of the paper is devoted to the proof of Theorems 1,2 and 3.

From now on $\Omega$ will be an annulus in the plane with boundary components $\Gamma_{\rm int},\Gamma_{\rm ext}$ which we assume convex and piecewise-smooth. We will write $\Omega=F\setminus\bar{G}$ where $F$ and $G$ are open, convex, with piecewise smooth boundary. In that case $\Gamma_{\rm int}=\partial G$ and $\Gamma_{\rm ext}=\partial F$.

Let $p$ be a point of $\partial G$ where $\partial G$ is not smooth ($p$ will then be called a vertex). The normal cone of $G$ at $p$ is the set

$$N_{G}(p)=\{x\in{\bf R}^{2}:\langle{x},{y-p}\rangle\leq 0,\quad\text{for all}\quad y\in G\}.$$

Then $N_{G}(p)$ is the closed exterior wedge bounded by the normal lines to the two smooth curves concurring at $p$, its boundary is the broken line depicted in the figure below. Call $\alpha_{p}$ its angle at $p$.

$\bullet\quad$We remark the obvious fact that $0<\alpha_{p}<\pi$.

We now define the minimum and maximum width in the piecewise-smooth case. These are defined in (9) and depicted in the Figure 3 below.

For a unit vector $v$ applied in $p$ and pointing inside $N_{G}(p)$ we let $\gamma_{p,v}(t)=p+tv$ denote the ray exiting $p$ in the direction $v$, and let $Q(p,v)$ be the intersection of $\gamma_{p,v}$ with $\Gamma_{\rm ext}=\partial F$. We define:

$$\left\{\begin{aligned} &\beta(p)=\inf_{v\in N_{G}(p)}d(p,Q(p,v))\\ &B(p)=\sup_{v\in N_{G}(p)}d(p,Q(p,v))\end{aligned}\right.$$

(9)

We notice that at a smooth point $q$ the cone at $q$ degenerates to the normal segment at $q$. Hence at a smooth point $q$ one has

$$\beta(q)=B(q).$$

We now define

$$\left\{\begin{aligned} &\beta(\Omega)=\inf\{\beta(p):p\in\partial G\}\\ &B(\Omega)=\sup\{\beta(p):p\in\partial G\}\end{aligned}\right.$$

(10)

$\beta(\Omega)$ and $B(\Omega)$ will be called the minimum width and, respectively, the maximum width of $\Omega$. We remark that when the two boundary components are smooth and parallel then $\beta=B$ and the ratio $\frac{\beta}{B}$ assumes its largest possible value, which is $1$.

As a first step in the proof of Theorem 1, we extend the inequality (2) to the piecewise-smooth case.

In this section we state the two main technical lemmas; the partition of the annulus $\Omega$ will be defined in Section 2.5.

So let $\Omega=F\setminus\bar{G}$ be an annulus as above and let $\beta\in(0,\beta(\Omega)]$. We consider the distance functions:

$$\rho_{1},\rho_{2}:F\to[0,\infty),$$

where $\rho_{1}(x)=d(x,G)$ and $\rho_{2}(x)=d(x,\partial F)$. Fix a parameter $\beta>0$. As $G$ is convex, with piecewise-smooth boundary, it is well-known that the equidistants $\{\rho_{1}=\beta\}$ are $C^{1}$-smooth curves. We say that the parameter $\beta$ is regular if the equidistant $\{\rho_{2}=\beta\}$ is a piecewise-smooth curve. Following Appendix 2 in [11], we know that the set of regular parameters has full measure in $(0,\beta(\Omega)]$; as a consequence, there exists a sequence of regular parameters $\{\beta_{j}\}\to\beta(\Omega)$ as $j\to\infty$.

$\bullet\quad$By using an obvious limiting procedure, from now on we take

$$\beta=\beta(\Omega)$$

and can assume that it is a regular parameter, so that $\rho_{2}=\beta$ is a piecewise-smooth curve.

We estimate the ratio $\frac{\beta}{B}$ of each piece as follows.

The proof of Lemma 6 and Lemma 7 involves rather simple geometric constructions, but there are some delicate points to take care of, and will be done in Section 2.5. In fact, these two lemmas make it possible to write $\Omega$ as a union of subset $\Omega_{k}$ such that the ratio $\frac{\beta(\Omega_{k})}{B(\Omega_{k})}$ is bounded below, which make the proof of Theorem 1 quite easy, as follows.

We use the partition $\{\Omega_{1},\dots,\Omega_{n}\}$ of Lemma 6. Let $A$ be a closed potential having flux $\Phi^{A}$ around the inner boundary curve $\partial G$; then, $A$ has the same flux around the inner component of $\Omega_{k}$, by Lemma 6a. By Proposition 4a there exists $k\in\{1,\dots,n\}$ such that

$$\lambda_{1}(\Omega,A)\geq\dfrac{1}{n}\lambda_{1}(\Omega_{k},A).$$

(12)

By Theorem 5 applied to $\Omega=\Omega_{k}$ we see:

$$\lambda_{1}(\Omega_{k},A)\geq\dfrac{4\pi^{2}}{\lvert{\partial F_{k}}\rvert^{2}}\dfrac{\beta(\Omega_{k})^{2}}{B(\Omega_{k})^{2}}d(\Phi^{A},{\bf Z})^{2}.$$

By b) of Lemma 7 we see:

$$\dfrac{\beta(\Omega_{k})^{2}}{B(\Omega_{k})^{2}}\geq\dfrac{1}{16}\dfrac{\lvert{F}\rvert^{2}}{D(F)^{4}}.$$

(13)

This, together with the inequality $\lvert{\partial F_{k}}\rvert\leq\lvert{\partial F}\rvert$, gives:

$$\lambda_{1}(\Omega_{k},A)\geq\dfrac{\pi^{2}}{4}\cdot\dfrac{\lvert{F}\rvert^{2}}{\lvert{\partial F}\rvert^{2}D(F)^{4}}\cdot d(\Phi^{A},{\bf Z})^{2}.$$

We insert this inequality in (12) and use the inequality $\frac{1}{n}\geq\frac{\beta}{2B(\Omega)}$ (see Lemma 6b) to conclude that

	$\displaystyle\lambda_{1}(\Omega,A)$	$\displaystyle\geq\frac{\beta}{2B(\Omega)}\lambda_{1}(\Omega_{k},A)$
		$\displaystyle\geq\dfrac{\pi^{2}}{8}\cdot\dfrac{\lvert{F}\rvert^{2}}{\lvert{\partial F}\rvert^{2}D(F)^{4}}\cdot\dfrac{\beta}{B(\Omega)}\cdot d(\Phi^{A},{\bf Z})^{2}.$

This proves part a) of Theorem 1.

If $\beta(\Omega)$ is less than the injectivity radius of $\partial F$ we proceed as before, using the lower bound $\frac{\beta(\Omega_{k})}{B(\Omega_{k})}\geq\frac{1}{\sqrt{2}}$ proved in Lemma 7c. We arrive easily at the inequality:

$$\lambda_{1}(\Omega,A)\geq\dfrac{\pi^{2}}{\lvert{\partial F}\rvert^{2}}\frac{\beta(\Omega)}{B(\Omega)}d(\Phi^{A},{\bf Z})^{2},$$

which is Theorem 1b).

We start by showing the partition on a particular example, see Figure 4 below. The initial domain is a triangle $F$ minus a disk $G$ and $\beta=\beta(\Omega)$. We draw the first three pieces $\Omega_{1},\Omega_{2},\Omega_{3}$ and then the last one, which is $\Omega_{6}$ and which coincides with the $\beta$-neighborhood of the exterior boundary $\partial F$ (this is always the case). Note that the pieces overlap, hence the partition is not disjoint.

We now proceed to construct the partition in general. Let then $\Omega$ be convex annulus $\Omega=F\setminus\bar{G}$ as above, and consider the distance functions:

$$\rho_{1},\rho_{2}:F\to[0,\infty),$$

where $\rho_{1}(x)=d(x,G)$ and $\rho_{2}(x)=d(x,\partial F)=d(x,F^{c})$.

$\bullet\quad$At step 1, we let $F_{1}=\{\rho_{1}<\beta\}$, $G_{1}=G$ and $\Omega_{1}=F_{1}\setminus\bar{G}_{1}$. That is, $\Omega_{1}$ is simply the subset of $F$ at distance less than $\beta$ to $G$.

$\bullet\quad$At step 2, we let $F_{2}=\{\rho_{1}<2\beta\}$ and $G_{2}=\{\rho_{1}<\beta\}\cap\{\rho_{2}>\beta\}$ and define $\Omega_{2}=F_{2}\setminus\bar{G}_{2}$.

$\bullet\quad$At the arbitrary step $k$, we let $F_{k}=\{\rho_{1}<k\beta\}$ and $G_{k}=\{\rho_{1}<(k-1)\beta\}\cap\{\rho_{2}>\beta\}$, and define:

$$\Omega_{k}=F_{k}\setminus\bar{G}_{k}.$$

Observe that for any choice of positive numbers $a,b$ the sets $\{\rho_{1}<a\}$ and $\{\rho_{2}<b\}$ are convex. Therefore, both $F_{k}$ and $G_{k}$ are convex; moreover $G_{k}\subset F_{k}$ and then $\Omega_{k}$ is an annulus. This proves part a) of Lemma 6.

For b) we first prove the following fact:

Fact. Let $n\doteq\Big{[}\dfrac{B(\Omega)}{\beta}\Big{]}$ (the smallest integer greater than or equal to $\frac{B(\Omega)}{\beta}$). Then $F_{n}=F$ and $G_{n}=\{\rho_{2}>\beta\}$. In particular, $\Omega_{n}=\{\rho_{2}<\beta\}$ and then, starting from $n$, the sequence $\Omega_{n}$ stabilizes: $\Omega_{n}=\Omega_{n+1}=\dots$.

For the proof we first observe that, from the definition of $B(\Omega)$, we have $F\subseteq\{\rho_{1}<B(\Omega)\}$; then, if we fix $n\geq\frac{B(\Omega)}{\beta}$ we have by definition $F\subseteq F_{n}$ hence $F=F_{n}$. To show that $G_{n}=\{\rho_{2}>\beta\}$ it is enough to show:

$$\{\rho_{2}>\beta\}\subseteq\{\rho_{1}<(n-1)\beta\}.$$

(14)

In fact, if not, there would be a point $x\in F$ such that $d(x,\partial F)=\rho_{2}(x)>\beta$ and $\rho_{1}(x)\geq(n-1)\beta$. Let $y\in\partial G$ be a point at minimum distance to $x$, and prolong the segment from $y$ to $x$ till it hits $\Gamma_{\rm ext}=\partial F$ at the point $z$. It is clear that then $d(y,z)=d(y,x)+d(x,z)>n\beta$. By definition of $B(\Omega)$ we then have:

$$B(\Omega)\geq d(y,z)>n\beta,$$

which contradicts the definition of $n$. Hence (14) holds.