6.4 The inner radius of nodal domains
6.4.6 An application: interior cone conditions
This shows that in order to bound ϕ−λ, it suffices to bound ϕ+λ. This finishes Step (2).
Step3: Controlled doubling exponent and conclusion:
Our aim is to be able to bound sup1
4Bϕ+λ in terms ofϕλ(x0), as that would give us control of the doubling exponent ofϕλon 18B. In other words, we wish to establish that
sup
1 4B
ϕ+λ ≤Cϕλ(x0), (6.113)
whereC is a constant independent of λ.
If X∩ 12B∩ {ϕ > 0} = ∅, then (6.113) follows immediately by definition. Otherwise, calling X0 :=X∩12B, let Ω0λrepresent another nodal domain on whichϕλis positive and which intersects X0. In other words, Ω0λ∩12B gives us a spike entering 12B whichϕλ is positive, and our aim is to obtain bounds on this spike.
Observe that (6.110) implies that the volume of the spike Ω0λ∩12B is small compared to 12B, and this allows us to invoke Theorem 6.4.6. We see that
21/ηϕλ(x0)&kϕλkL∞(M) (by hypothesis (6.105))
≥sup
Ω0λ
ϕλ≥21/η sup
Ω0λ∩12B
ϕλ≥21/η sup
Ω0λ∩14B
ϕλ(by applying Theorem 6.4.6).
Now (6.113) follows, which implies that the growth is controlled in the ball 18B, that is, β1/8(ϕλ) =sup1
4B|ϕλ| sup1
8B|ϕλ| ≤c1, (6.114)
wherec1 depends on (M, g) and not on0or λ(in particular, not onr0, η).
Now, we bring in the asymmetry estimate (Proposition 6.4.1), which, together with (6.114), tells us that
|{ϕλ<0} ∩14B|
|14B| ≥c2, (6.115)
wherec2is a constant depending only onc1and (M, g). But selecting the constantC2to be smaller thanc2we see that (6.115) contradicts (6.110). Hence, we obtain a contradiction with the fact that the function ϕλ vanishes inside 14B.
Finally, this proves that with the initial choice of parameters, there is an inscribed ball of radius
r0
4√
λ inside Ωλ. By construction, we had thatr0∼η(n−1)(n−2)2n =ηβ(n).
Combined with the inner radius estimates in [Man08a], this proves the claim of Theorem 6.4.4.
Furthermore, in dimension 2, the nodal lines form an equiangular system at a singular point of the nodal set.
Setting dimM ≥3, we discuss the question whether at the singular points of the nodal setNϕ, the nodal set can have arbitrarily small opening angles, or even “cusp”-like situations, or the nodal set has to self-intersect “sufficiently transversally”. We observe that in dimensionsn≥3 the nodal sets satisfies an appropriate “interior cone condition”, and give an estimate on the opening angle of such a cone in terms of the eigenvalue λ.
Now, in order to properly state or interpret such a result, one needs to define the concept of
“opening angle” in dimensionsn≥3. We start by defining precisely the notion of tangent directions in our setting.
Definition 6.4.4. LetΩλ be a nodal domain andx∈∂Ωλ, which means thatϕλ(x) = 0. Consider a sequence xn ∈ Nϕ such that xn → x. Let us assume that in normal coordinates around x, xn =exp (rnvn), where rn are positive real numbers, and vn ∈S(TxM), the unit sphere in TxM. Then, we define the space of tangent directions at x, denoted bySxNϕ as
SxNϕ={v∈S(TxM) :v= limvn, wherexn∈Nϕ, xn→x}. (6.116) We note that there are more well-studied variants of the above definition, for example, as due to Clarke or Bouligand (for more details, see [Roc79]). With that in place, we now give the following definition of “opening angle”.
Definition 6.4.5. We say that the nodal set Nϕ satisfies an interior cone condition with opening angleα atx∈Nϕ, if any connected component ofS(TxM)\ SxNϕ has an inscribed ball of radius
&α.
Now we have the following:
Theorem 6.4.8. When dim M = 3, the nodal set Nϕ satisfies an interior cone condition with angle& √1λ. When dimM = 4, Nϕ satisfies an interior cone condition with angle& λ7/81 . Lastly, when dimM ≥5,Nϕ satisfies an interior cone condition with angle& 1λ.
We will use Bers scaling of eigenfunctions near zeros (see [Ber55]). We quote the version as appears in [Zel08], Section 3.11.
Theorem 6.4.9(Bers). Assume thatϕλvanishes to orderkatx0. Letϕλ(x) =ϕk(x) +ϕk+1(x) + ...denote the Taylor expansion ofϕλinto homogeneous terms in normal coordinatesxcentered at x0. Then ϕκ(x)is a Euclidean harmonic homogeneous polynomial of degreek.
We also recall the inradius estimate for real analytic metrics from Theorem 6.4.2.
Since the statement of Theorem 6.4.2 is at first glance asymptotic in nature, we need to note that a nodal domain corresponding toλwill still satisfy inrad (Ωλ)≥ cλ3 for some constantc3 even for small λ. This follows from the inradius estimates of Mangoubi in [Man08a], which hold for all frequencies. Consequently, we can assume that every nodal domain Ω onSn corresponding to the spherical harmonicϕk(x), as in Theorem 6.4.9, has inradius& λ1.
Proof of Theorem 6.4.8. We observe that Theorem 6.4.2 applies to spherical harmonics, and in particular the function exp∗(ϕk), restricted toS(Tx0M), whereϕk(x) is the homogeneous harmonic polynomial given by Theorem 6.4.9. Also, a nodal domain for any spherical harmonic on S2 (respectively,S3) corresponding to eigenvalueλhas inradius∼ √1λ (respectively,&λ7/81 ).
With that in place, it suffices to prove that
Sx0Nϕ⊆ Sx0Nϕk. (6.117)
By definition, v ∈ Sx0Nϕ if there exists a sequence xn ∈Nϕ such thatxn →x0, xn = exp(rnvn), wherern are positive real numbers andvn ∈S(Tx0M), andvn→v.
This gives us,
0 =ϕλ(xn) =ϕλ(rnexpvn)
=rnkϕk(expvn) + X
m>k
rnmϕm(expvn)
=ϕk(expvn) +X
m>k
rnm−kϕm(expvn)
→ϕk(expv), asn→ ∞. Observing thatϕk(x) is homogeneous, this proves (6.117).
Chapter 7
Obstacles
7.1 Formulation and background
In this Chapter, we consider the problem of placing of an obstacle in a domain so as to maximize the fundamental frequency of the complement of the obstacle. To be more precise, let Ω⊂Rn be a bounded domain, and let Dbe another bounded domain referred to as ”obstacle”. The problem is to determine the optimal translatex+Dso that the fundamental Dirichlet Laplacian eigenvalue λ1(Ω\(x+D)) is maximized/minimized.
In case the obstacleD is a ball, physical intuition suggests that for sufficiently regular domains and sufficiently small balls, Ω,λ1(Ω\Br(x)) will be maximized whenx=x0, a point of maximum of the ground state Dirichlet eigenfunction φλ1 of Ω. Heuristically, such maximum points x0 seem to be situated deeply in Ω, hence removing a ball aroundx0should be an optimal way of truncating the lowest possible frequency. Our methods give equally good results for Schr¨odinger operators on a large class of bounded domains sitting inside Riemannian manifolds (see the remarks at the end of Section 7.2). In terms of exposition, we follow our work in [GM17a].
The following well-known result of Harrell-Kr¨oger-Kurata treats the case when Ω satisfies convexity and symmetry conditions:
Theorem 7.1.1([HKK06]). LetΩbe a convex domain inRnandBa ball contained inΩ. Assume that Ωis symmetric with respect to some hyperplane H. Then,
(a) at the maximizing position,B is centered onH, and (b) at the minimizing position, B touches the boundary of Ω.
The last result of Harrell-Kr¨oger-Kurata seems to work under rather strong symmetry assumption.
We also recall that the proof of Harrell-Kr¨oger-Kurata proceeds via a moving planes method which essentially measures the derivative of λ1(Ω\B) when B is shifted in a normal direction to the hyperplane.
There does not seem to be any result in the literature treating domains without symmetry or convexity properties.
In the following discussion, we consider bounded domains Ω⊂Rn which satisfy an asymmetry assumption in the following sense:
Definition 7.1.1. A bounded domain Ω ⊂Rn is said to satisfy the asymmetry assumption with coefficientα(or Ωisα-asymmetric) if for all x∈∂Ω, and all r0>0,
|Br0(x)\Ω|
|Br0(x)| ≥α. (7.1)
This condition seems to have been introduced in [Hay78]. We also recall that theα-asymmetry property was utilized by D. Mangoubi in order to obtain inradius bounds for Laplacian nodal domains (cf. Section 6.4 and also [Man08a]) as nodal domains are asymmetric withα= λ(n−1)/2C .
From our perspective, the notion of asymmetry is useful as it basically rules out narrow ”spikes”
(i.e. with relatively small volume) entering deeply into Ω. For example, let us also observe that convex domains trivially satisfy our asymmetry assumption with coefficientα= 12.