No hot spots

Euclidean Triangles have no hot spots

Abstract.

We show that a second Neumann eigenfunction of a Euclidean triangle has at most one (non-vertex) critical point , and if exists, then it is a non-degenerate critical point of Morse index . Using this we deduce that

  1. the extremal values of are only achieved at a vertex of the triangle, and

  2. a generic acute triangle has exactly one (non-vertex) critical point and that each obtuse triangle has no (non-vertex) critical points.

This settles the ‘hot spots’ conjecture for triangles in the plane.

Key words and phrases:
Hot spots, Laplace operator, Neumann eigenfunctions
1991 Mathematics Subject Classification:
35P05, 35B38, 35J05, 35J25, 58J50
The work of C.J. is partially supported by a Simons collaboration grant.

1. Introduction

Let be a domain in Euclidean space with Lipschitz boundary. The second Neumann eigenvalue, , is the smallest positive number such that there exists a not identically zero, smooth function that satisfies

(1)

where denotes the outward pointing unit normal vector field defined at the smooth points of . A function that satisfies (1) will be called a second Neumann eigenfunction for , or simply a -eigenfunction.

One variant of the ‘hot spots’ conjecture, first proposed by J. Rauch at a conference in 1974,1 asserts that a second Neumann eigenfunction attains its extrema at the boundary. The main result of this paper implies the hot spots conjecture for triangles in the plane.

Theorem 1.1.

If is a second Neumann eigenfunction for a Euclidean triangle , then has at most one critical point.2 Moreover, if has a critical point , then lies in a side of and is a nondegenerate critical point with Morse index equal to .

In Theorem 12.4, we show that if is a generic acute triangle, then has exactly one critical point, and that if is an obtuse triangle, then has no critical points. Earlier, Bañuelos and Burdzy showed that if is obtuse, then has no interior maximum, and, in particular, the maximum and minimum values of are achieved at the acute vertices [Bnl-Brd99]. We extend the latter statement to all triangles (see Theorem 12.1). Unlike [Bnl-Brd99], our proof of Theorem 1.1 does not rely on probabilistic techniques.

For a brief history and various formulations of the ‘hot spots’ conjecture, we encourage the reader to consult [Bnl-Brd99]. We provide some highlights. The first positive result towards this conjecture was due to Kawohl [Kwl85] who showed that the conjecture holds for cylinders in any Euclidean space. Burdzy and Werner in [Brd-Wrn99] (and later Burdzy in [Brd05]) showed that the conjecture fails for domains with two (and one) holes. In the paper [Brd05] Burdzy made two separate (‘hot spot’) conjectures for ‘convex’ and ‘simply connected’ domains. We believe that the conjecture is true for all convex domains in the plane.

The conjecture has been settled for certain convex domains with symmetry. In 1999, under certain technical assumptions, Bañuelos and Burdzy [Bnl-Brd99] were able to handle domains with a line of symmetry. A year later Jerison and Nadirashvili [Jrs-Ndr00] proved the conjecture for domains with two lines of symmetry. In a different direction, building on the work in [Bnl-Brd99], Atar and Burdzy [Atr-Brd04] proved the conjecture for lip domains (a domain bounded by the graphs of two Lipschitz functions with Lipschitz constant 1). In 2012, the hot spots conjecture for acute triangles became a ‘polymath project’ [Polymath]. In 2015 Siudeja [Sdj15] proved the conjecture for acute triangles with at least one angle less than by sharpening the ideas developed by Miyamoto in [Mym09, Mym13]. Notably, in the same paper, Siudeja proved that the second Neumann eigenvalue of an acute triangle is simple unless is an equilateral triangle. An earlier theorem of Atar and Burdzy [Atr-Brd04] gave that the second Neumann eigenvalue of each obtuse and right triangle is simple.

Our approach to the conjecture differs from most of the previous approaches (but has some features in common with the approach in [Jrs-Ndr00]). For each acute or obtuse triangle, , we consider a family of triangles that joins to a right isosceles triangle . Using the simplicity of (due to [Atr-Brd04], [Mym13] and [Sdj15]) we then consider a family of second Neumann eigenfunctions associated to .3 Because is the right isosceles triangle, the function is explicitly known up to a constant, and a straightforward computation shows that has no critical points (see equation (21)). Therefore, if were to have a critical point, then it would have to somehow ‘disappear’ as tends to 1. Each nondengenerate critical can not disappear immediately, that is, it is ‘stable’. On the other hand, a degenerate critical point can instantaneously disappear, that is, it could be ‘unstable’. Thus, as varies from to , either a critical point of converges to a vertex or is or becomes degenerate and then disappears. Understanding the first case, among the last two possibilities, is more or less straightforward, and we do it by studying the expansion of in terms of Bessel functions near each vertex. Understanding the second case is more complicated. One particular reason for this complication is that disappearance of this type probably does occur for perturbations of general domains.

The study of how eigenvalues and eigenfunctions vary under perturbations of the domain is a classical topic (see for example [Kato]). Jerison and Nadirashvilli [Jrs-Ndr00] considered one-parameter families of domains with two axes of symmetry and studied how the nodal lines of the directional derivatives of the associated eigenfunctions varied. In particular, they used the fact that each constant vector field commutes with the Laplacian, and hence if is an eigenfunction, then is also an eigenfunction with the same eigenvalue. The eigenfunctions were also used in [Sdj15] and implicitly in [Bnl-Brd99] and [Atr-Brd04]

In the current paper, we consider the vector field , called the rotational vector field, that corresponds to the counter-clockwise rotational flow about a point . To be precise if , then

We will call the angular derivative of about . Each rotational vector field commutes with the Laplacian, and hence the angular derivative is an eigenfunction. By studying the nodal sets of where is a vertex of the triangle, one finds that if has an interior critical point, then also has a critical point on each side of the triangle (see Corollary 6.2). Moreover, we show that each of these three critical points is stable under perturbation even though might not be stable (see Proposition 9.4). We also use the nodal sets of both and , where is parallel to the side of , to show that, although a degenerate critical point of might not be stable under perturbation, there are at least two other critical points that are stable under perturbation (see Proposition 9.5).

Outline of the paper

In §2, we recall Cheng’s [Chn76] theorem concerning the structure of the nodal set of an eigenfunction on surfaces. From a result of Lojasiewicz [Ljs59] it follows that the critical set of each eigenfunction is a disjoint union of isolated points and analytic one-dimensional manifolds. In §3, we consider domains obtained from a triangle via reflecting about its sides. By applying Cheng’s structure theorem to the extension of an eigenfunction to these extended polygonal domains, we obtain a qualitative result concerning nodal arcs whose endpoints lie in a side of the triangle. In §4, we consider the Bessel expansion of a Neumann eigenfunction on a sector. Using the radial and angular derivatives of this expansion, we obtain a qualitative description of the critical set of a Neumann eigenfunction on a sector. We use this lower estimate in §5 to prove that the critical set of a second Neumann eigenfunction on a triangle is finite. There we also (re)prove the fact that the nodal set of is a simple arc, and use this fact to obtain information about the first two Bessel coefficients of at the vertices of . For example, we deduce that can vanish at only one vertex of . In §6, we study the nodal set of both the angular derivatives, , about vertices and the directional derivatives, , parallel to an edge . We show that each component of each of these nodal sets is a finite tree, and use this to obtain information about the critical set of . For example, we show that if has an interior critical point then it has at least three more critical points, one critical point per side (Corollary 6.2), and if has a degenerate critical point on a side , then has a critical point on a side distinct from (Theorem 6.5).

In §8, we begin the proof of Theorem 1.1. Given an obtuse or (non-equilateral) acute triangle , we consider a ‘straight line path’ of triangles that joins to a right isosceles triangle and an associated path of second Neumann eigenfunctions. In §8, we suppose converges to and consider the accumulation points of a sequence where each is a critical points of . Using the Bessel expansion of , we find that if each lies in the interior of , then a vertex is not an accumulation point of . We also show that if each lies in a side and a vertex is an accumulation point of , then there does not exist a sequence of critical points lying in a distinct side that has as an accumulation point.

In §9, we address the issue of the stability of critical points. We regard a critical point of as ‘stable’ if for each neighborhood of the function has a critical point in for sufficiently close to . Non-degenerate critical points are stable, but, in general, degenerate critical points are not. Nonetheless, we use the results of §6 to show that if is a degenerate critical point of , then has at least two stable critical points. In §10, we use the existence of these two stable critical points to show that, if has an interior critical point, then also has at least two critical points for each that is near . In contrast, the eigenfunction for the right isosceles triangle has no critical points, and thus, to prove Theorem 1.1 for acute and obtuse triangles, it suffices to show that the number of critical points can not drop from two to zero in the limit as tends to one. This is accomplished by using the results of §8 and certain elementary properties of .

To make the exposition of the proof of Theorem 1.1 easier, we use the known simplicity of the second Neumann eigenvalue [Bnl-Brd99] [Atr-Brd04] [Mym13], [Sdj15]. However, we indicate in §11 how to avoid this assumption.

In §12, we use the topology of the nodal sets of the extension of to the double of the triangle to show that has a critical point if and only if each vertex is an isolated local extremum of . In particular, if is associated to an acute triangle, then has a critical point if and only if does not vanish at any of the vertices. In the final part of §12, we consider the parameter space of all labeled triangles up to homothety. Using analytic perturbation theory and Hartog’s separate analytic theorem we deduce that the Bessel coefficients of a second Neumann eigenfunction (at a labeled vertex) can be thought of (in a suitable sense) as analytic functions on a dense open subset of . Using this fact, we deduce that a generic acute triangle has exactly one critical point and obtuse triangles have no critical points.

Notation and terminology.

For notational convenience, we will regard the Euclidean plane as the complex plane. That is, we will use to represent a point in the plane. In particular, and , and if then and . We will also use to denote the set of such that , and to denote the interior of a set . For us, a Laplace eigenfunction is a smooth real valued solution to the equation where and . We will sometimes call such a solution a -eigenfunction.

Acknowledgments.

We thank Neal Coleman for producing contour plots of eigenfunctions in triangles. In particular, he created a very inspirational animation of the ‘straight-line’ family of triangles joining a triangle with labeled angles to a triangle with labeled angles . See https://youtu.be/bO50jFOxCAw. He created these contour plots with his ‘fe.py’ python script [Clm16]. We also thank David Jerison and Bartlomiej Siudeja for comments on the first version of the paper.

2. The nodal set and the critical set of an eigenfunction

Let be an open set, and let be an eigenfunction of the Laplacian. In this section, we recall some facts about the nodal set and the set, , of critical points of . The intersection is the set of nodal critical points.

The following is a special case of the stratification of real-analytic sets due to Lojasiewicz [Ljs59]. An elementary proof can be found in the proof of Proposition 5 in [Otl-Rss09].

Lemma 2.1.

Let be an open subset of . If is a real-analytic function, then each has a neighborhood such that is either equal to or is homeomorphic to a properly embedded finite graph. Moreover, if , then is a real-analytic arc.

Because the Laplacian is a constant coefficient elliptic operator, the eigenfunction is real-analytic function. Therefore, it follows from Lemma 2.1 that is a locally finite graph whose vertices are the nodal critical points, and the complement of these vertices is a disjoint union of real-analytic loops and arcs. Cheng observed [Chn76] that (in dimension 2) the nodal set has a special structure in a neighborhood of each nodal critical point.

Lemma 2.2 (Theorem 2.5 in [Chn76]).

Let be an eigenfunction of the Laplacian on an open set . If is a nodal critical point, then there exist a neighborhood of , a positive integer , a real number , and simple arcs , such that

  1. ,

  2. equals , and

  3. for each , the arc is tangent at to the line .

Remark 2.3.

Arcs satisfying condition (3) of Lemma 2.2 are called equiangular.

Sketch of proof.

Without loss of generality . The Taylor series of about may be regarded as a sum of homogeneous polynomials of degree in and . Because is a nodal critical point, and vanish identically. Since is an eigenfunction and maps homogeneous polynomials of degree to homogeneous polynomials of degree , we have . In particular, if is the smallest such that , then . Thus, where is a harmonic polynomial of degree at least and denotes a sum of terms of degree at least . The restriction of the harmonic polynomial to the unit circle centered at is a Laplace eigenfunction with eigenvalue , and so since is homogeneous, the nodal set of equals the union of lines for some . One obtains the claim by applying the method of [Kuo69]. See Lemma 2.4 in [Chn76]. ∎

As a consequence of Lemma 2.2, the nodal set is the union of loops and proper4 arcs. We will call these the Cheng curves of .

We shall be interested in whether certain Cheng arcs cross a line or not. To make this precise, we note the following.

Lemma 2.4.

Let be a Cheng curve in and let be an intersection point of and a line . There exists an open neighborhood of and a parameterization of such that and either

  1. the sets and lie in different components of ,

  2. the sets and lie in the same component of or

  3. the curve lies in the component of that contains .

In case (1), we say that the curve crosses the line .

Proof.

The restriction of to is a real-analytic function on . We have case (3) if and only if this restriction vanishes identically on the component containing . If it it does not vanish identically, then there exists a neighborhood of such that contains no zeros of the restriction other than . Choose a parameterization of so that and do not intersect . ∎

The set, , of critical points has the following description parallel to that of the nodal set .

Proposition 2.5.

Let be a Laplace eigenfunction on an open set . Each connected component of is either

  1. an isolated point,

  2. a proper real-analytic arc, or

  3. a real-analytic curve that is homeomorphic to a circle.

Proof.

The function is analytic, and hence by Lemma 2.1 each critical point is either isolated or lies in a component of that is a locally finite graph.

Let be a component of the graph . If for some , then since is connected and on , it would follow that . By Lemma 2.2, the set of nodal critical points is discrete, and hence would consist of an isolated point.

If , either or . Without loss of generality, we may assume that , and hence . Therefore, the analytic implicit function theorem provides a neighborhood of such that is a real-analytic arc . The set lies in .

By Lemma 2.1, the set is either finite, and hence is an isolated point, or is a proper finite graph. In the latter case . Since is arbitrary, the component is a real-analytic 1-manifold (without boundary). If is compact, then is homeomorphic to a circle. Otherwise, there exists a possibly infinite open interval and a real-analytic unit speed parameterization . Since is closed in , the map is proper. ∎

3. Eigenfunctions on triangles, kites, and hexagons

In this section we consider eigenfunctions on the triangle that satisfy Neumann conditions along at least one of the sides of . Let be a side of , and let denote the reflection across the line containing . Following [Sdj15], we define the kite to be the closed set . If is an eigenfunction of the Laplacian that satisfies Neumann conditions along , then extends uniquely to a real-analytic Neumann eigenfunction on the kite such that . Note that whenever we refer to the nodal set of an extended eigenfunction, we are speaking of the nodal set in the extended domain.

If is an eigenfunction that satisfies Neumann conditions on all three sides, then we will find it useful to reflect about all three sides simultaneously. If some angle of is greater than , then one might not be able to extend to the union of the three kites unambiguously. But one may use a ‘smaller’ extension. For example, the bisectors of each angle of the triangle meet at the centroid to form a tripod. This tripod divides the triangle into three smaller triangles each of which contains exactly one edge of . Reflect each of these smaller triangles about the corresponding edge to obtain a ‘hexagon’ containing . The Neumann eigenfunction on extends uniquely via the reflection principle to a Laplace eigenfunction on .

Let be an eigenfunction of the Laplacian on the interior of that extends continuously to the vertices of . In this article will equal or where is either a constant or rotational vector field. By Lemma 2.2, the nodal set of is a union of curves where each curve is either homeomorphic to a circle (a ‘loop’) or is a proper arc. Recall that each such curve is called a Cheng curve of .

Definition 3.1.

Let be a Cheng curve of . The closure of a component of the intersection will be called a maximal subset of the nodal set of the restriction of to .

The nodal set of the restriction is a union of maximal subsets. Each maximal subset in the nodal set of is either a point, a loop5, or a arc with distinct endpoints in . Each intersection of such loops/arcs is equiangular (see Remark 2.3). If a maximal subset is homeomorphic to an interval, then we will call it a maximal arc.

If a maximal subset consists of a single point, then this point lies in a side of . Indeed, the nodal set of an eigenfunction defined on an open set has no isolated points. For the same reason, if satisfies Neumann or Dirichlet conditions along a side of , then contains no singleton maximal subsets. In particular, the nodal set of the Neumann eigenfunction is a union of maximal loops and maximal arcs. Each vertex of the graph is thus either a critical point of , an endpoint of a maximal arc, or an isolated point of .

The following should be compared to Lemma 5 in [Sdj15].

Lemma 3.2.

Let be a triangle. Let be an eigenfunction on that satisfies Neumann conditions along the side . If a piecewise smooth arc in has both endpoints in , then the eigenvalue of is strictly greater than the second Neumann eigenvalue of .

Proof.

The maximal arc and the side together bound a topological disc . Define by setting if and otherwise. The function satisfies Neumann conditions along and Dirichlet conditions along the other two sides of . Hence the eigenvalue, , of is larger than the first eigenvalue of the mixed eigenvalue problem on corresponding to Neumann conditions on and Dirichlet conditions on the other two sides. In turn, by Theorem 3.1 in [Ltr-Rhl17], the first eigenvalue of the mixed problem is greater than the second Neumann eigenvalue of . ∎

4. Neumann eigenfunctions on sectors

Let be a sector of angle and radius , that is

In this section, is a (real) eigenfunction of the Laplacian on with eigenvalue that satisfies Neumann boundary conditions along the boundary edges corresponding to respectively. (We impose no conditions on the circle of radius .) We will use the expansion of in Bessel functions near the ‘vertex’ , to derive information about both the nodal set and the critical set of .

Separation of variables leads to the following expansion valid near :

(2)

Here and denotes the Bessel function of the first kind of order [Lbv72]. The series converges uniformly on compact sets that miss the origin. The Bessel function has the expansion [Lbv72]

where is the Gamma function. In particular, for each , there exists an entire function so that .6 Note that none of the Taylor coefficients of vanish. In particular, neither nor vanishes in a neighborhood of for each . With this notation, the expansion in (2) takes a more compact form:

(3)

where .

We will be interested in the level set, , that contains the vertex . In particular, if , then is the nodal set of .

Lemma 4.1.

There exists a neighborhood of such that either equals or equals the union of real-analytic arcs such that the pairwise intersection of and equals for each .

Proof.

By expanding each , the expansion in (3) becomes

(4)

where each is nonzero. We have . Let and let . Then

(5)

where is a real-analytic function and both and are of order as tends to zero for some . The claim follows from the implicit function theorem. ∎

We will require more specialized information about the level sets that contain a vertex of a triangle when the vertex angle is acute or obtuse.

Lemma 4.2.

If the angle , then there exists a neighborhood of such that

  1. if , then equals , and

  2. if and , then is a simple arc containing .

If , then there exists a neighborhood of such that

  1. if , then is a simple arc containing , and

  2. if and , then

If , then there exists a neighborhood of such that consists of at least two arcs.

Proof.

Suppose . If , then defined in Lemma 4.1 equals and . In particular, the trigonometric polynomial appearing on the right hand side of (5) is a constant and hence . On the other hand, if and , then and . In this case, the trigonometric polynomial of the right hand side of (5) equals , and hence is a simple arc.

Suppose . If , then and . Thus, the trigonometric polynomial equals , and is an arc. On the other hand, if and , then the trigonometric polynomial is a constant and hence .

Finally, if , then each term in the trigonometric polynomial in (5) is the product of a constant and where and . Such a function has at least two roots. ∎

Proposition 4.3.

If is not an integer multiple of , then there exists a deleted neighborhood of that contains no critical points of . If , then there exists a neighborhood of such that is either empty, equals , or equals exactly one edge of the sector.

Remark 4.4.

The conditions on are necessary. For example, on the square we have the Neumann eigenfunction . In this case, the set lies in the critical set of .

Proof.

The point is a critical point of if and only if both the radial derivative and the angular derivative vanish at . Let be the first nonzero coefficient in the Bessel expansion in (3). By differentiating term-by-term we obtain

(6)

and

(7)

In particular,

(8)

and

(9)

where represents a function defined in a neighborhood of that is bounded by a constant times .

Suppose that and . Then since , we find from (8) that . It follows that there exists so that

(10)

Suppose that . If , then , and so from (9) we find that . It follows that there exists so that

Therefore, if , there exists such that if then and can not both be zero.

If , then the term associated to in (7) might not be dominant and so (9) might not be useful. Which term is dominant depends on the value of .

If , then the term associated to is dominant, and thus does not vanish for small . If , then the term associated to is dominant, and we find that there exists so that for some . Comparison with (10) where then gives that and can not both vanish near .

If , then since satisfies Neumann conditions along the edges, we may use the reflection principle to extend to a smooth eigenfunction on the disk . By Proposition 2.5, if lies in the critical locus of , then there exists a disk neighborhood of zero such that or is a real-analytic arc . Because the extended eigenfunction is invariant under reflection across both the real and imaginary axes, the arc is also invariant under these reflections and hence lies either in the real or imaginary axis. ∎

Remark 4.5.

If , then the sector is a half-disk. One can apply the reflection principle to extend to the disk. Using Proposition 2.5, we find that if is a critical point, then there exists a neighborhood of such that is either , equals the real-axis, or is an arc that is orthogonal to the real-axis.

Remark 4.6.

If , then there exists , such that if and , then is not a critical point of . Indeed, for each , defines an analytic function on , and hence from (6) we have

where . Thus, if is a critical point and then there exists such that

(11)

and therefore .

5. A second Neumann eigenfunction on a Euclidean triangle

In this section, is a Euclidean triangle, and is a second Neumann eigenfunction for .

We will use the following well-known fact many times in the sequel.

Lemma 5.1.

Let be a subset of with piecewise smooth boundary, and let that satisfies Dirichlet boundary conditions on , that is . Then the Rayleigh quotient . In particular, if itself is a -eigenfunction on with Dirichlet boundary condition, then .

Proof.

By the variational characterization of the first Dirichlet eigenvalue we have . By the domain monotonicity of the first Dirichlet eigenvalue, and by a result of Polya [Ply52] we have , giving the first assertion. ∎

The following fact is also well-known.

Theorem 5.2.

The nodal set of consists of one simple maximal arc.

Proof.

By Lemma 2.2, the nodal set is a collection of loops and maximal arcs. Lemma 5.1 implies that there are no loops. By Courant’s nodal domain theorem, the complement has exactly two components. The claim follows. ∎

If is a vertex of the triangle , then an -neighborhood of can be identified with a subset of a sector. For each vertex , we consider the Bessel expansion of about , and we let denote the associated Bessel coefficient.

Corollary 5.3.

Let be a vertex of . The first two Bessel coefficients, and , can not both equal zero.

Proof.

If both were both zero, then by Lemma 4.2, there would exist (at least) two arcs in that emanate from the vertex. They could not form a loop by Lemma 5.1, and so they would have to be distinct, but this would contradict Theorem 5.2. ∎

Corollary 5.4.

If and are two distinct vertices of , then and can not both equal zero.

Proof.

Suppose to the contrary that and are both zero. Then by Corollary 5.3, the coefficients and are both nonzero. Thus, by Lemma 4.2, there would exist an arc in emanating from and an arc in emanating from . By Theorem 5.2 these arcs would belong to the same maximal arc in that joins and . This would contradict Lemma 3.2. ∎

The following is a consequence of a more general result of [Ndr86], but it follows easily from the previous corollary.

Corollary 5.5.

The dimension of the space of second Neumann eigenfunctions is at most two.

Proof.

Define the linear map by where and are distinct vertices of . By Corollary 5.4, the map has no kernel, and so the dimension of is at most two. ∎

Proposition 5.6.

The critical set of a second Neumann eigenfunction is finite.

Proof.

The Neumann eigenfunction extends via reflection to an eigenfunction on the interior of the ‘hexagon’ described in §3. By Proposition 2.5, each component of is either an isolated point, a proper analytic arc, or an analytic loop. It follows that each component of the critical set of is either an analytic arc with points in the boundary of , a loop in , or an isolated point in where is the set of vertices.

If were a loop, then each directional derivative, for example , would be a Dirichlet eigenfunction on the region bounded by the loop, contradicting Lemma 5.1.

If were an arc, then the endpoints of the arc lie in the union of two sides. If is the rotational vector field about the common vertex of these two sides, then is a Dirichlet eigenfunction on a subdomain of . This would contradict Lemma 5.1.

Thus each component of is an isolated point in . If each vertex angle is not equal to , then Proposition 4.3 implies that there is a neighborhood of the set, , that contains no critical points. Therefore is finite if is not a right triangle.

If is a right triangle, then Proposition 4.3 gives that either a deleted neighborhood of contains no critical points or one of the sides is a component of . In the former case, the preceding argument still applies. The latter case is impossible. Indeed, the other endpoint of the side is a vertex with angle strictly less than , contradicting Proposition 4.3. ∎

6. Derivatives of a second Neumann eigenfunction

In this section, is a second Neumann eigenfunction for a triangle . Here, we consider the nodal sets of the angular and directional derivatives of .

By ‘directional derivative’ we mean the result of applying a (real) constant vector field . Each such vector field commutes with the Laplacian and so if is an eigenfunction of the Laplacian, then is also an eigenfunction with the same eigenvalue. We are particularly interested in the unit vector field, , that is parallel to a side of a triangle such that a counterclockwise rotation of points into the half-plane containing . We will let denote the unit vector field that is outward normal to the side . Note that satisfies Neumann conditions if and only if for each side of .

By ‘angular derivative’ we mean the result of applying the rotational vector field that corresponds to the counter-clockwise rotational flow about a point . To be precise if , then

The vector field commutes with the Laplacian, and so if is a Laplace eigenfunction, then is also an eigenfunction with the same eigenvalue. We are particularly interested in the case where is a vertex of a triangle.

Recall that a tree is a simply connected graph. The degree of a vertex is the number of edges that contain the vertex. By the interior of a side of the triangle , we will mean the complement where are the vertices of . We will denote the interior with .

According to §3, the nodal sets of both and are locally finite graphs whose vertex set consists of the critical points of , endpoints of maximal arcs, and isolated points in the boundary of . We will now show that each of these graphs is finite and each component is a tree.

Lemma 6.1.

Let be a vertex of and let denote the side opposite to . The nodal set of is a finite disjoint union of finite trees, and it contains the sides adjacent to . If the nodal set of intersects the interior of , then the nodal set has a degree 1 vertex that lies in the interior of . Each point that lies in the intersection of and is a critical point of .

Proof.

The simple connectedness of the nodal set follows from Lemma 5.1.

If is a side that is adjacent to , then the restrictions of the vector fields and to agree up to a non-zero factor. Thus, since vanishes along , so does .

On the other hand, for each in the side opposite to , the vector is independent of the vector . Hence, if belongs to the interior of , then is a critical point of . Therefore by Lemma 5.6, the intersection is finite.

Suppose that lies in the intersection of and the interior of . Since is simply connected, the component of that contains is a maximal arc that has two distinct endpoints. Since the sides adjacent to are contained in , one of these endpoints lies in the interior of the side opposite to . This endpoint is a critical point of .

In sum, the set is the union of the sides adjacent to and the maximal arcs that have at least one endpoint in the interior of , and each such endpoint is a critical point of . Since there are only finitely many critical points and the degree of each vertex of is finite, there are finitely many maximal arcs. It follows that the set is a finite disjoint union of finite trees.

Each finite tree contains at least two degree 1 vertices. Let be a (tree) component of that intersects the interior of . If does not contain the union, , of the sides adjacent to , then each degree 1 vertex of lies in . Otherwise, note that the closure of is a finite union of trees, and let be a component that intersects the interior. Exactly one vertex of lies in , and hence at least one degree 1 vertex of lies in . ∎

Corollary 6.2.

If has a critical point that lies in the interior of , then for each vertex of , the nodal set of has a degree 1 vertex that lies in the interior of the side opposite to . In particular, if has a critical point that lies in the interior of , then has at least three more critical points each lying in a distinct side of .

Proof.

If is a critical of , then for each vertex of . Lemma 6.1 implies the claim. ∎

Lemma 6.3.

Let be a side of . The nodal set of is a finite union of finite trees. If a maximal arc of intersects the interior of , then one endpoint of the arc lies in , and if is not the vertex opposite to , then is a critical point of . If the nodal set of intersects the interior of , then the nodal set has a degree 1 vertex that lies in .

Proof.

Lemma 5.1 implies that the nodal set is simply connected.

Since satisfies Neumann conditions, the function satisfies Neumann conditions along . If , then there exists a maximal arc of containing that has distinct endpoints. The endpoints can not both lie in as a consequence of Lemma 3.2. Hence at least one endpoint lies in .

If is a side of that meets at an angle equal to , then . Thus, it follows from Lemma 3.2, that if one of the endpoints of the latter maximal arc lies in then the other can not lie on and hence lies on . If meets at an angle not equal to , then at each , the vectors and are independent. In particular is a critical point of .

Thus, if is a maximal arc that intersects the interior of , then at least one endpoint of is a critical point that lies in . By Lemma 5.6, the set of such points is finite. Each vertex of has finite degree and so the number of maximal arcs in is finite. It follows that is a finite disjoint union of finite trees.

The remainder of the argument is similar to that given at the end of the proof of Lemma 6.1. ∎

Lemma 6.4.

Let be a side of . The intersection equals the set of critical points of that lie in , and is hence finite. Each point in is an endpoint of at least one maximal arc of that intersects .

Proof.

The first assertion follows from the fact that and are independent. By Lemma 5.6, the function has only finitely many critical points.

Since satisfies Neumann conditions along , we may extend uniquely to an eigenfunction on the interior of the kite that is invariant under the reflection associated to . Since is parallel to , we find that is also invariant under , and hence the nodal set is also invariant. No Cheng arc of equals , and therefore there exists a maximal arc that intersects the interior of . ∎

The following theorem plays a prominent role in the proof Theorem 1.1.

Theorem 6.5.

Let be a side of . If has a degenerate critical point that lies in and does not have a critical point that lies in the interior of , then either

  1. for each of the vertices adjacent to , the nodal set of has a degree 1 vertex that belongs to the edge opposite to , or

  2. the nodal set of has a degree 1 vertex that belongs to the interior of a side distinct from , and the nodal set of has a degree 1 vertex that belongs to the interior of a side distinct from .

Proof.

Without loss of generality, the critical point is located at the origin, and the edge lies in the real axis. Let be the extension of to the interior of the kite obtained by reflecting about . The real-analytic Taylor expansion at has the form

(12)

where indicates a sum of terms in and of order at least . Since is a Neumann eigenfunction along , we have . Therefore, since by assumption is a degenerate critical point, either or .

The case leads to alternative (1). Indeed, if , then from (12) we find that the angular derivative of about an endpoint of equals

It follows that is a nodal critical point of . By Lemma 2.2, at least two Cheng curves of intersect at and the intersection is equiangular. In particular, since one of these arcs is , some other curve is transverse to at and hence intersects . Hence by Lemma 6.1, the nodal set of has a degree 1 vertex that lies in the interior of the side opposite to . (See Figure 1.) Letting , the two endpoints of , we obtain alternative (1).

The case leads to alternative (2). Indeed, in this case, from (12) we find that . Thus at least two Cheng curves of meet at . Note that is invariant under the reflection about , and hence these Cheng curves are also invariant. By Lemma 6.4 none of these Cheng curves is a subset of . It follows that each curve intersects the interior of . By Lemma 6.3, each of two of the corresponding maximal arcs, , has an endpoint . If is a vertex of , then the other endpoint, , lies in the interior of a side . If and lie in the interiors of distinct sides , then choose and . If and lie in the interior of the same side , then by relabeling if necessary, we may assume that and the vertex opposite to are separated by , and we choose . In fact, in each case the curve separates from the vertex opposite to . By Lemma 6.3, is a critical point of , and moreover, there exists a degree vertex of that lies in the interior of . (See Figure 1.)

Since is a critical point of , by Lemma 6.4, there exists a maximal arc of that intersects the interior of and has as an endpoint. The other endpoint of cannot be the vertex opposite to . Indeed, since the vectors and are independent at each , an intersection point would be a critical point. By considering a subtree containing the arc one finds a degree 1 vertex of that lies in the interior of a side distinct from . (See Figure 1.) ∎

Figure 1. Some cases in Theorem 6.5. In the figure on the far left, a possible nodal set of is described. Possible nodal sets of are described in the second and fourth pictures from the left, and possible nodal sets of are described in the third and fifth pictures.
Corollary 6.6.

If has exactly one critical point , then is a nondegenerate critical point.

Proof.

By Corollary 6.2, the point lies in . By Theorem 6.5, the point is nondegenerate. ∎

We will also use the next two results in the proof of Theorem 1.1, but Lemma 6.8 will not be used in the acute case.

Proposition 6.7.

Let be a side of . If contains at least two critical points of , then has a third critical point that lies in .

Proof.

Let and be critical points of that lie on . By Lemma 6.4, there exists a maximal arc of that intersects the interior of and has as an endpoint in . The other endpoints of