A functional H Ω of the level sets

[20, 35]. This enables us to give a proof of Theorem 2.1.2 for the first time, thus providing a complete answer to the problem.

We will include the background results from [35] leading up to and including the proof of Theorem 2.1.1, both for completeness’ sake, and because the exposi-tion would be rather difficult to follow otherwise. We include some of the more important or illustrative proofs from there in this chapter; the others, especially the more technical proofs, have been reproduced in Appendix B. Finally, as in [35], we will set the proof up so it works for Dirichlet boundary conditions as well, thus giving a new proof of the sharpness of the inequality in Theorem 2.1.3. Note however that our method only works for Wiener (or Dirichlet) regular bounded domains; see Remark 2.2.2. The new material in this chapter has been published in [39]; since publication the results have been generalised and developed in other directions in [22, 23, 31].

2.2. A functional H_Ω of the level sets

In order to cover the Robin and Dirichlet cases simultaneously, in this section we will consider the eigenvalue problem

(2.2.1)

−∆u = λu in Ω, u = 0 on Γ0,

∂u

∂ν + αu = 0 on Γ₁,

on a given bounded domain Ω ⊂ R^N, where Γ₀, Γ₁ are disjoint open and closed subsets of ∂Ω with Γ0 ∪ Γ1 = ∂Ω (see also (1.2.1)). Here we will assume either Ω is of class C², or else Ω is Wiener regular (see Remark 2.2.2) and Γ1 = ∅. If Γ1 = ∅, then we have a pure Dirichlet problem, while if Γ⁰ = ∅, then we have a pure Robin problem. We will also assume that α > 0 is a constant, although for this section and the next we could assume without loss of generality that (for example) α ∈ C¹(Γ1). Finally, we may assume without loss of generality that Ω is connected. Indeed, for disconnected Ω, given the Faber-Krahn inequality for connected domains it is immediate that λ1(Ω) > λ1(B). This follows from applying the Faber-Krahn inequality to each connected component of Ω (see Remark 1.3.2) and then using strict monotonicity of λ₁(B) with respect to the volume of B (see Lemma 1.3.7).

2.2. A functional HΩof the level sets 23

We introduce the following notation. For open sets U ⊂ Ω we denote the interior and exterior boundaries by ∂iU := ∂U∩Ω and ∂eU := ∂U∩∂Ω, respectively (see also (A1.1)). We will be interested in the case where the subsets U are the level sets of the first eigenfunction ψ of (2.2.1). Recall that λ1(Ω) is simple, and that ψ can be chosen to be strictly positive in Ω (see Remark 1.3.3). We will normalise ψ ≥ 0 so that kψk∞= 1, and denote the level sets of ψ by

(2.2.2) Ut :={x ∈ Ω : ψ(x) > t}

and the level surfaces by

(2.2.3) S_t:={x ∈ Ω : ψ(x) = t},

where t ∈ [0, 1]. Note that since ψ ∈ C(Ω) ∩ C^∞(Ω) by Theorem 1.2.8, Sard’s lemma [68, Theorem 3.1.3] implies St is, locally, a C^∞ (N − 1)-manifold inside Ω for almost every t ∈ (0, 1), although a priori the intersection with ∂Ω could be nasty. Moreover the level sets U_t are open. In particular S_t must coincide with the interior boundary ∂iUt of Ut for almost all t∈ (0, 1) with respect to Lebesgue measure on (0, 1) (cf. also the comments around (A4.6)).

The principal reason for assuming that Ω is of class C² is so that we have the extra regularity of the eigenfunction from Theorem 1.2.8, namely that

ψ ∈ C^∞(Ω)∩ C¹(Ω)∩ W^2,p(Ω)

for all p ∈ (1, ∞). If Γ1 = ∅, then Ut is compactly contained in Ω since the sets {x ∈ Ω : ψ(x) = 0} ⊇ ∂Ω and Ut = {x ∈ Ω : ψ(x) ≥ t} are compact and disjoint. In this case, by Sard’s lemma, St = ∂Ut is a C^∞ manifold for almost every t∈ (0, 1). We also set

m := min

x∈Ωψ(x)≥ 0.

By Theorem 1.3.1(iv) and Remark 1.3.3, ψ(x) > 0 for all x ∈ Γ1, and ψ attains its minimum m on ∂Ω; if Γ0 =∅ then m > 0, while clearly otherwise m = 0. Finally, we observe that St=∅ if t /∈ (m, 1], and Ut∩ Γ0 =∅ for all t ∈ (m, 1).

We next recall the following rather technical result from [35] concerning the behaviour of the level surfaces St, which will needed in the sequel.

Lemma 2.2.1 ([35], Lemma 2.3). The following are true.

(i) The function t7→ σ(St) is in L¹((0,∞)).

2.2. A functional HΩof the level sets 24

(ii) The St are of class C^∞ and the Ut are Lipschitz for almost all t∈ (m, 1).

(iii) If Γ1 6= ∅, then there exist c > 0 and t1 ∈ (m, 1) such that σ(St)≤ cσ(∂Ω) for all t ∈ (m, t¹].

The proof can be found in Appendix B. We observe that (i) uses the coarea formula, (ii) uses Sard’s lemma, and (ii) and (iii) both require Ω to be “reasonably smooth” if Γ1 6= ∅. The situation is different if Γ1 =∅.

Remark 2.2.2. (i) If Γ1 = ∅, the only regularity assumption on ∂Ω we need is that ψ ∈ C(Ω) in order to ensure all level sets U^t are compactly contained in Ω. This assumption on ψ is equivalent to the assumption that Ω is Wiener (or Dirichlet) regular (see, e.g., [10]). We remark that this is a far weaker condition than Lipschitz regularity.

(ii) Note also that the actual inequality in Theorem 2.1.3 can be obtained for all domains Ω of finite volume, not necessarily bounded, using a standard perturbation argument as in [35, Section 4]. The sharpness of the inequality can be obtained for all bounded domains Ω, but it requires a different method from the one above. A proof using symmetrisation is included in [39, Section 4]. For arbitrary bounded domains the sharpness only holds up to sets of capacity zero, since removing a set of capacity zero from a domain Ω will not increase its eigenvalues (this is well known; see for example [24, Section 2] and the references therein).

With this background material in mind, we are ready to look at the method behind the proof of Theorems 2.1.1 and 2.1.2. The key is the following functional.

If U ⊂ Ω is open with U ∩ Γ0 = ∅, and ϕ ∈ C(Ω) is non-negative, then as in [20, 35], we let

(2.2.4) HΩ(U, ϕ) := 1

|U|

Z

∂iU

ϕ dσ + Z

∂eU

α dσ− Z

U|ϕ|²dx .

Since ϕ is continuous, each of these integrals makes sense; moreover, we will be working with restricted choices of U and ϕ for which the last integral is finite.

Thus HΩ will always be well-defined, although HΩ(U, ϕ) = ∞ is possible. More precisely, the subsets U will be the level sets Ut of ψ; in particular, for almost all t the ∂iU will be the level surfaces St. If Γ0 =∅, we also restrict our choice of test functions ϕ; see Section 2.3.

The main reason why the functional H_Ω is useful is that it can be used to obtain an estimate of the first eigenvalue λ1(Ω) of (2.2.1) (see Section 2.3). For

2.2. A functional HΩof the level sets 25

this, the function |∇ψ|/ψ = |∇(ln ψ)| will play an important role. The following proposition motivates the definition of HΩ. Although this was already proved in [35, Proposition 2.1], we include the straightforward but informative proof.

Proposition 2.2.3. Let ψ be a positive first eigenfunction of the problem (2.2.1).

Then

2.2. A functional HΩof the level sets 26

Remark 2.2.4. Another way of looking at the above argument is via the weak formulation. That is, we use 1/ψ as a test function in the weak form Qα(ψ, v) =

More generally, to obtain HΩ for a general level set Ut, we do the same thing for the problem

since inserting 1/ψ into the weak equation for this problem will clearly yield (2.2.5) whenever Ut is Lipschitz (i.e. whenever everything in the above expression makes sense). This does not by itself weaken the smoothness requirement on Ω; however this has subsequently been achieved in [22] by combining this idea with a cut-off argument.

We will use Proposition 2.2.3 to obtain a characterisation of λ1(Ω) in terms of HΩ for arbitrary ϕ∈ C(Ω) non-negative. For this we need another technical result from [35]; a proof is in Appendix B. Given ϕ∈ C(Ω) non-negative we set

(2.2.6) w := ϕ− |∇ψ|

Lemma 2.2.5 ([35], Lemma 3.3). Suppose that ϕ ∈ C(Ω) is non-negative such that ϕ∈ L¹(U) for every open set U ⊂ Ω with U ⊂ Ω ∪ Γ1. Then F is absolutely

2.2. A functional HΩof the level sets 27

The proof makes use of the coarea formula (see Theorem A4.6). Note that absolutely continuous functions are differentiable almost everywhere (for a defini-tion and discussion of them, see [99, Chapter 7]). We can now give the important characterisation of λ₁(Ω) mentioned earlier.

Theorem 2.2.6. Let ϕ∈ C(Ω) be non-negative. Then (2.2.8) HΩ(Ut, ϕ) = λ1(Ω)− 1

Proof. We use the definition of w to obtain an expression for the difference be-tween HΩ(Ut, ϕ) and HΩ(Ut,|∇ψ|/ψ). First,

Now fix t ∈ (m, 1) such that the results of Section 2.2 hold. We apply the coarea formula (Theorem A4.6), valid for any non-negative measurable, not necessarily integrable, function to obtain characterisation (2.2.5) of λ₁(Ω), we see that

H_Ω(U_t, ϕ) = 1

Since this holds for almost every t ∈ (m, 1), this gives us the desired result.