Asymptotics of sloshing eigenvalues for a triangular prism
AASYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM
JULIEN MAYRAND, CHARLES SENÉCAL, SIMON ST-AMANT
Abstract.
We consider the three-dimensional sloshing problem on a triangular prism whose angleswith the sloshing surface are of the form π q , where q is an integer. We are interested in finding atwo-term asymptotic expansion of the eigenvalue counting function. When both angles are π , wecompute the exact value of the second term. As for the general case, we conjecture an asymptoticexpansion by constructing quasimodes for the problem and computing the counting function of therelated quasi-eigenvalues. These quasimodes come from solutions of the sloping beach problem andcorrespond to two kinds of waves, edge waves and surface waves. We show that the quasi-eigenvaluesare exponentially close to real eigenvalues of the sloshing problem. The asymptotic expansion oftheir counting function is closely related to a lattice counting problem inside a perturbed ellipsewhere the perturbation is in a sense random. The contribution of the angles can then be detectedthrough that perturbation. Introduction
The Steklov and sloshing problems.
Let Ω ⊂ R n be a bounded domain with boundary Γ and let ρ ∈ L ∞ (Γ , R ) be a non-negative weight function. The Steklov problem with weight ρ consists of finding all solutions u ∈ H (Ω) and σ ∈ R of the problem(1) (cid:40) ∆ u = 0 in Ω ,∂ ν u = σρu on Γ , where ∆ = (cid:80) ni =1 ∂ x i and ∂ ν denotes the exterior normal derivative on the boundary. The classicalSteklov problem consists in having ρ ≡ on Γ .Our main interest is the sloshing problem. Given a partition of the boundary Γ = Γ N (cid:116) Γ S ,the sloshing problem consists of solving (1) with ρ ≡ on Γ N and ρ ≡ on Γ S . It is a mixedSteklov-Neumann boundary problem describing the oscillations of an ideal fluid in a tank shapedlike Ω with walls Γ N and free surface (or sloshing surface) Γ S . The admissible values of σ are calledthe sloshing eigenvalues.1.2. Our problem.
Let Σ ⊂ R be a triangle with a side S = [0 , L ] ×{ } of length L making angles α at (0 , and β at ( L, with the other sides. We denote the union of those two other sides by W . Given M > , we consider the sloshing problem on the rectangular prism Ω = Σ × [0 , M ] ⊂ R with sloshing surface Γ S = S × [0 , M ] and walls Γ N = ( W × [0 , M ]) ∪ (Σ × { } ) ∪ (Σ × { M } ) . All this notation is summarized in Figure 1 where the sloshing surface is shaded in grey.The sloshing problem on Ω consists of finding functions Φ : Ω → C such that(2) ∆Φ = 0 in Ω ,∂ ν Φ = 0 on Γ N ,∂ ν Φ = σ Φ on Γ S . for some σ ∈ R . It is a mixed Steklov-Neumann boundary problem describing the oscillations of anideal fluid in a tank shaped like Ω . The sloshing eigenvalues correspond to the eigenvalues of the a r X i v : . [ m a t h . SP ] J u l JULIEN MAYRAND, CHARLES SENÉCAL, SIMON ST-AMANT xy z O Σ WS α β LM Figure 1.
Example of domain Ω with α = β = π .Dirichlet-to-Neumann map DN : H / (Γ S ) → H − / (Γ S ) which maps a function u to ∂ ν ˜ u where ˜ u is the solution to ∆˜ u = 0 in Ω ,∂ ν ˜ u = 0 on Γ N , ˜ u = u on Γ S . It is a positive semi-definite self-adjoint operator with compact resolvent. As such, its eigenvaluesform a discrete sequence σ < σ ≤ σ ≤ · · · (cid:37) ∞ accumulating at infinity. By separating variables (see [7, Lemma 2.1]), it is sufficient to considerfunctions of the form Φ( x, y, z ) = cos( λ n z ) ϕ ( x, y ) with λ n = nπM where ϕ : Σ → R satisfies(3) ∆ ϕ = λ n ϕ in Σ ,∂ ν ϕ = 0 on W ,∂ ν ϕ = σϕ on S . We are interested in the asymptotic expansion of the eigenvalue counting function N ( σ ) := { j ∈ N : σ j < σ } . From [1], we know that N ( σ ) = LM π σ + o ( σ ) . This asymptotic does not capture the contribution from the angles α and β . Our goal is to find asuitable second term in the asymptotic expansion for N ( σ ) which reveals how both angles affect thecounting function. We will be more particularly interested in the case where α = π q and β = π r forsome integers q and r greater or equal to , but not both . Remark 1.1.
The case α = β = π obviously does not result in a triangular prism and would actuallygive rise to an unbounded domain. However, the asymptotic behavior of the sloshing eigenvaluesshould only depend on a neighborhood of the sloshing surface. This intuition is supported by the SYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM 3 following computation. Consider the cuboid
Ω = [0 , L ] × [0 , R ] × [0 , M ] ⊂ R with the sloshingsurface corresponding to y = R . As above, we can separate variables to get eigenfunctions of theform cos( λ n z ) ϕ ( x, y ) with ϕ satisfying (3). We can then separate variables again in the x directionto get eigenfunctions of the form Φ( x, y, z ) = cos (cid:16) mπL x (cid:17) cos (cid:16) nπM z (cid:17) Y ( y ) where m and n are non-negative integers and the function Y satisfies Y (cid:48) (0) = 0 , Y (cid:48) ( R ) = σY ( R ) and Y (cid:48)(cid:48) Y = (cid:16) mπL (cid:17) + (cid:16) nπM (cid:17) =: µ . It follows that Y ( y ) = cosh( µy ) and the eigenvalue is given by σ = µ tanh( µR ) . As m or n getlarge, so does µ , and tanh( µR ) converges to exponentially fast. Hence, σ = µ + O ( µe − µR ) andthe eigenvalues barely depend on R . The eigenvalue counting function is then given by N ( σ ) = (cid:26) ( m, n ) ∈ N : (cid:16) mπσL (cid:17) + (cid:16) nπσM (cid:17) < (cid:27) + o ( σ ) = LM π σ + L + M π σ + o ( σ ) . This last expression comes from estimates on the Gauss circle problem (see [16] for example).Therefore, the asymptotic behavior of N ( σ ) does not depend on R . Remark 1.2.
We expect that the asymptotic behavior of the sloshing eigenvalues should onlydepend on a neighborhood of the sloshing surface. Therefore, the results we will show on theasymptotic behavior of N ( σ ) should also be valid in the more general case where W is a piecewisesmooth curve y = w ( x ) with w (0) = w ( L ) = 0 , w ( x ) < for x ∈ (0 , L ) , and making the sameangles α and β with S .1.3. Motivation.
The sloshing problem has its origins in the theory of hydrodynamics (see [13,Chapter 9] for example). It describes the oscillations of an ideal fluid on the surface of a container,such as coffee in a cup. Modern results and references on the sloshing problem can be found in [10]and [11].There has been recent interest into the Steklov problem (1), see [9] for a survey on the problem.The Steklov eigenvalues correspond to the eigenvalues of the Dirichlet-to-Neumann map which isoften referred to as the voltage-to-current map. It is very closely related to the Calderòn problem[5] upon which lies electrical impedance tomography, used in geophysical and medical imaging.If ∂ Ω and ρ are smooth, the Dirichlet-to-Neumann operator is a pseudodifferential operator andone can use pseudodifferential techniques to study its spectrum [8, 12, 19, 20]. However, whenever ∂ Ω is not smooth (in the presence of corners for example), those techniques fail and other approacheshave to be considered. The simplest example of Ω without a smooth-boundary is a cuboid in R n .The eigenvalue counting function on cuboids has been studied in [7] where they showed that itadmits a two-term asymptotic where the second term accounts for the n − dimensional facets ofthe cuboid, e.g. the length of the edges in a regular cube. However, in the case of a cuboid, allthe angles between the facets are the same right angles. Changing the angles should change theasymptotic and that is what we wish to quantify.The problem we are considering stems from the work of Levitin, Parnovski, Polterovich and Sherin [14] and [15]. In both papers, their goal is to understand how angles inside a two dimensionalcurvilinear polygon affect its Steklov or sloshing eigenvalues. They started off by considering thesame triangles Σ as we described in 1.2. Their goal was then to solve(4) ∆ u = 0 in Σ ,∂ ν u = 0 on W ,∂ ν u = σu on S . JULIEN MAYRAND, CHARLES SENÉCAL, SIMON ST-AMANT
This problem is exactly like the problem (3) with n = 0 . They were able to show the following. Theorem 1.3 (Levitin, Parnovski, Polterovich, Sher [14], 2019) . Suppose that < α ≤ β < π .Then the following asymptotic expansion holds for the eigenvalues of problem (4) as k → ∞ : σ k L = π (cid:18) k − (cid:19) − π (cid:18) α + 1 β (cid:19) + o (1) . A key idea of their proof was to reduce the problem to angles of the form π q for q ∈ N , whichare refered to as exceptional angles. They then used domain monotonicity to show the result forarbitrary angles α and β by bounding them from above and below by exceptional angles. Consideringthese exceptional angles allowed them to compute explicitly solutions from the sloping beach problememanating from each corner which they glued together to obtain approximate solutions of (4) calledquasimodes. Through careful analysis of the quasimodes, they were able to show that the relatedquasi-eigenvalues were close to real eigenvalues of problem (4) and approximated all of them.We now aim to generalize their approach to three dimensions. By separating variables, we canbring everything back to two dimensions, but we are now solving for solutions of the Helmholtzequation with different eigenvalues λ n rather than for harmonic functions.1.4. Main results.
Our first result concerns the case where α = β = π and is obtained by findingexplicitly the eigenfunctions. Theorem 1.4.
The eigenvalue counting function of problem (2) with α = β = π is given by N ( σ ) = LM π σ + L + M (2 √ π σ + o ( σ ) . For other values of α and β , we were not able to find the eigenfunctions explicitly and it probablyis unfeasible. Hence, we have to resort to new methods. Our idea is to construct quasimodesthat are approximate solutions of problem (2). More specifically, our quasimodes will satisfy theeigenvalue condition on the sloshing surface, but rather than satisfy the Neumann condition on thewalls, the normal derivative will decay exponentially with respect to their eigenvalue σ . Hence,the quasimodes will be very close to being eigenfunctions and we should expect the error betweenquasi-eigenvalues and real eigenvalues of the problem to converge to zero as they get large. We willuse two kinds of quasimodes that we refer to as edge waves and surface waves. Their constructionis presented in Section 3. Let N e ( σ ) and N s ( σ ) be the counting functions for the eigenvalues of theedge waves and surface waves respectively. Our main results then concern the asymptotic expansionof those counting functions. Before stating them, we need to introduce some quantities.Let α = π q and β = π r . Define θ α ( t ) = − q − (cid:88) j =1 arctan (cid:32) √ − t sin jπq − cos jπq (cid:33) . and define similarly θ β by substituting q by r . Furthermore, let ν α,β = qr mod 2 and κ α,β be if q and r share the same parity, and otherwise. Then, we show the following two theorems. Theorem 1.5.
The counting function N e ( σ ) for the edge waves quasi-eigenvalues satisfies thefollowing asymptotic expansion: N e ( σ ) = ν α,β M σπ + (cid:98) q − (cid:99) (cid:88) m =0 M σπ sin(2 m + 1) α + (cid:98) r − (cid:99) (cid:88) (cid:96) =0 M σπ sin(2 (cid:96) + 1) β + O (1) . SYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM 5
Theorem 1.6.
The counting function N s ( σ ) for the surface waves quasi-eigenvalues satisfies thefollowing asymptotic expansion: N s ( σ ) = LM π σ + L − M π σ + κ α,β Mπ σ + M σπ (cid:18)(cid:90) [ θ α ( t ) + θ β ( t )] d t (cid:19) + o ( σ ) . Ideally, these quasi-eigenvalues would correspond to the real eigenvalues of the sloshing problem.We will show that for every quasi-eigenvalue, there is a sloshing eigenvalue exponentially close toit. Indeed, if we denote by { ˜ σ j } j ∈ N the set of our quasi-eigenvalues arranged in ascending order,then Lemma 5.1 implies the following. Lemma 1.7.
There exist positive constants C and c such that for every j ∈ N , there exists k ( j ) ∈ N such that (cid:12)(cid:12) ˜ σ j − σ k ( j ) (cid:12)(cid:12) ≤ Ce − c ˜ σ j . Hence, by showing that all but finitely many values of k ( j ) can be chosen distinctly, we can showthat N ( σ ) is bounded from below by the sum of our quasi-eigenvalue counting functions. Theorem 1.8.
The eigenvalue counting function N ( σ ) of problem (2) satisfies N ( σ ) ≥ N e ( σ ) + N s ( σ ) + o ( σ ) . However, we will not be able to show that there is a quasi-eigenvalue close to every real eigenvalueof the sloshing problem, which would show that N ( σ ) ≤ N e ( σ ) + N s ( σ ) + o ( σ ) . This leads us theconjecture the following. Conjecture 1.9.
The eigenvalue counting function N ( σ ) of problem (2) is given by N ( σ ) = N e ( σ ) + N s ( σ ) + o ( σ ) . Note that when α = β = π or α = β = π , this coincides with what we got in Remark 1.1 andwhat we show in Theorem 1.4. Although we are not able to prove Conjecture 1.9 for other angles,we provide numerical evidence supporting it in Section 5.3. As mentioned above, this conjecturehinges on showing that there is a quasi-eigenvalue next to each sloshing eigenvalue. This motivatesthe next definition and our second conjecture. Definition 1.10.
We say that the sequence of quasi-eigenvalues ˜ σ j is asymptotically complete if wecan choose the function k in Lemma 1.7 in a way that there exists integers N > and J ∈ Z , suchthat for any j > N , k ( j ) = j + J .This definition is inspired by the similar definition in [14], but without the “quasi-frequency gap”condition. Conjecture 1.11.
The set of all edge wave and surface wave quasi-eigenvalues is asymptoticallycomplete.
Note that Conjecture 1.11 implies Conjecture 1.9. We also support Conjecture 1.11 with numericalevidence in Section 5.3. A priori, the integer J in the definition of asymptotic completeness can beof any sign. Moreover, it appears from our numerical experiments that the larger q and r are, thelarger J gets. Finding the specific value of J is a separate issue, but it is clear that it depends onboth angles.Both our conjectures are only valid for angles of the form π q . At the moment, we are unable todeal with arbitrary angles, see Section 5.2. JULIEN MAYRAND, CHARLES SENÉCAL, SIMON ST-AMANT
Our approach.
Firstly, in Section 2, we compute explicitly the eigenfunctions and eigenvaluesfor the case where α = β = π . From those computations, we show Theorem 1.4. Then, in Section3, using solutions coming from the theory of the sloping beach problem, we construct quasimodesfor any angles α = π q and β = π r . These solutions arise in two forms that we refer to as edgewaves and surface waves, corresponding to the discrete and continuous parts of the spectrum of thesloping beach problem (see [21]). Using these quasimodes, we find suitable asymptotic formulas for N e and N s in Section 4, showing Theorems 1.5 and 1.6. Counting the eigenvalues coming fromedge wave solutions is straightforward. However, counting the eigenvalues coming from surfacewave solutions is more involved and we reduce the problem to that of counting integer points in arandomly perturbed ellipse. We discuss the theory of quasimodes and show Theorem 1.8 in Section5, as well as provide numerical evidence of Conjectures 1.9 and 1.11.1.6. Acknowledgments.
The research of J.M. and C.S. was supported by NSERC’s USRA, andwas done as part of an intership at Université de Montréal, under the supervision of Iosif Polterovich.The research of S.St-A. was supported by NSERC’s CGS-M and FRQNT’s M.Sc. scholarship (B1X).This work is part of his M.Sc. studies at the Université de Montréal, under the supervision of IosifPolterovich. Authors would like to thank Iosif Polterovich for useful discussions and guidance. Theywould like to thank Zeev Rudnick for the proof of Lemma 4.4 and introducing them to the theory ofexponential sums. S.St-A. would also like to thank Thomas Davignon and Alexis Leroux-Lapierrefor useful discussions, as well as Jean Lagacé, Michael Levitin, Leonid Parnovski and David Sherfor their comments. 2.
Explicit computation of the case α = β = π Consider the cuboid ˜Ω = [ − L/ , L/ × [0 , M ] . xy z π/ π/ O LML
Figure 2.
Reflections of Ω along Γ N to get ˜Ω .Let ˜Γ S ⊂ ∂ ˜Ω denote the four faces of the cuboid with area LM and let ˜Γ N ⊂ ∂ ˜Ω denote the twofaces of the cuboid with area L . If Φ : Ω → R is a solution of (2), then the function ˜Φ : ˜Ω → R SYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM 7 obtained by reflecting evenly Φ along a rectangular part of Γ N three times satisfies(5) ∆ ˜Φ = 0 in ˜Ω ,∂ ν ˜Φ = 0 on ˜Γ N ,∂ ν ˜Φ = σ ˜Φ on ˜Γ S . We illlustrate these reflections in Figure 2 (note that we changed the position of the origin O fromFigure 1). Conversely, if ˜Φ is a solution of (5) that is symmetric along both planes spanned bythe rectangular parts of Γ N , then Φ = ˜Φ | Ω is a solution of (2). Therefore, solving (2) is equivalentto finding solutions with even symmetries along these planes. In other words, the functions mustbe invariant under the change of variables ( x, y ) (cid:55)→ ( y, x ) and ( x, y ) (cid:55)→ ( − y, − x ) . Finding suchsolutions is much easier since we can separate variables completely.Let λ n = nπM for n ∈ N . The corresponding eigenfunctions then take the form ˜Φ( x, y, z ) = ϕ ( x, y ) cos( λ n z ) where ϕ ( x, y ) is given by one of the functions in Table 1. One can check that all these eigenfunctionssatisfy ϕ ( x, y ) = ϕ ( y, x ) = ϕ ( − y, − x ) .Eigenfunction ϕ Conditions on χ and n Eigenvalue cosh (cid:16) λ n √ x (cid:17) cosh (cid:16) λ n √ y (cid:17) n ≥ λ n √ tanh (cid:16) λ n √ L (cid:17) sinh (cid:16) λ n √ x (cid:17) sinh (cid:16) λ n √ y (cid:17) n > λ n √ coth (cid:16) λ n √ L (cid:17) cos( χx ) cosh( (cid:112) χ + λ n y )+ cos( χy ) cosh( (cid:112) χ + λ n x ) n ≥ − χ tan χL (cid:112) χ + λ n tanh (cid:18)(cid:112) χ + λ n L (cid:19) − χ tan χL sin( χx ) sinh( (cid:112) χ + λ n y )+ sin( χy ) sinh( (cid:112) χ + λ n x ) n ≥ χ cot χL (cid:112) χ + λ n coth (cid:18)(cid:112) χ + λ n L (cid:19) χ cot χL xy n = 0 L Table 1.
Eigenfunctions ϕ ( x, y ) obtained by separation of variables that are sym-metric with respect to y = x and y = − x .Let N ( i ) ( σ ) be the number of eigenvalues of problem 2 smaller than σ corresponding to eigen-functions in the i -th line of Table 1 for i = 1 , . . . , . First, since there is only one function of type , N (5) ( σ ) = O (1) . Second, since the hyperbolic tangents and cotangents quickly converge to , wehave N (1) ( σ ) = N (2) ( σ ) = √ Mπ σ + O (1) . We can rewrite the third condition on χ and n as(6) χ = πL (cid:18) − π arctan (cid:20)(cid:112) λ n /χ ) tanh (cid:18)(cid:112) χ + λ n L (cid:19)(cid:21) + 2 m (cid:19) JULIEN MAYRAND, CHARLES SENÉCAL, SIMON ST-AMANT for m ∈ N . Similarly, the fourth condition is given by(7) χ = πL (cid:18) − π arctan (cid:20)(cid:112) λ n /χ ) coth (cid:18)(cid:112) χ + λ n L (cid:19)(cid:21) + (2 m + 1) (cid:19) where again m ∈ N . We only consider the positive solutions of χ as the negative solutions give riseto the same eigenfunctions. When m = 0 , equation (6) admits no solution χ > . Notice that thehyperbolic tangents and cotangents quickly converge to as σ = (cid:112) χ + λ n + O ( e − σ ) gets big, andhence the solutions of both equations (6) and (7) are exponentially close to the solutions of χ = πL (cid:18) m − π arctan (cid:112) λ n /χ ) (cid:19) for m ∈ N . The eigenvalues are given by σ = (cid:112) χ + λ n + O ( e − σ ) and so σ = (cid:16) m − π arctan (cid:112) λ n /χ ) (cid:17) πL + (cid:16) nπM (cid:17) + O ( e − σ ) . Moreover, we have arctan (cid:112) λ n /χ ) = − arctan (cid:112) − ( λ n /σ ) + π O ( e − σ ) . By plugging this relation into the previous equation and including the π into the integer m , itfollows that the eigenvalues σ of type and are exponentially close to the solutions of(8) σ = (cid:16) m + π arctan (cid:112) − ( λ n /σ ) (cid:17) πL + (cid:16) nπM (cid:17) for m ≥ and n ≥ . In Section 4, we show how to count the number of solutions of such anequation. Theorem 1.4 then follows from those calculations.It is important to note the behavior of the eigenfunctions in Table 1. We can ignore the singularsolution xy since it doesn’t contribute significantly to N ( σ ) . The first two functions are concen-trated in the corners of the square [ − L/ , L/ . Hence, the corresponding solutions Φ on Ω areconcentrated on the edges of the sloshing surface that have length M . It makes sense to call suchsolutions edge waves . On the other hand, the third and fourth solutions are concentrated on theedge of the square [ − L/ , L/ where they oscillate. Therefore, the corresponding solutions Φ on Ω oscillate on the whole sloshing surface, but vanish fast inside Ω . In contrast to the edge waves,we refer to those solutions as surface waves .Hence, in order to approximate solutions on a domain Ω with angles α = π q and β = π r , we haveto consider both kinds of waves. In the next section, we show how to construct these solutions foreach type of wave. 3. Construction of quasimodes
In order to approximate solutions of the sloshing problem, we are going to glue together solu-tions of a similar problem emanating from both corners. The functions we obtain are not exactlyeigenfunctions for our problem. Nonetheless, they give rise to eigenvalues that should be close tothe actual eigenvalues. We refer to them as quasi-eigenvalues. We discuss the theory of quasimodesin Section 5. The functions we use arise from the solutions of the sloping beach problem whichhas both discrete and continuous spectrum (see [21] and [4]). We construct quasimodes comingfrom both parts of the spectrum. We refer to the solutions corresponding to the discrete part ofthe spectrum as edge waves since they will generate quasimodes concentrated on the edges of theprism Ω . In analogy, we refer to the solutions corresponding to the continuous part of the spectrum SYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM 9 as surface waves since the resulting quasimodes will oscillate on the whole sloshing surface Γ S anddecay exponentially inside Ω . Lemmas 3.1 and 3.6 will confirm the behaviors of the edge wave andsurface wave quasimodes respectively.Note that although the spectrum corresponding to surface waves is continuous, the resultingquasi-eigenvalues will be discrete, since we will get “gluing” conditions in order for our resultingapproximate solutions to be sufficiently smooth.3.1. Sloping beach problem.
Consider the angular sector S α = {− α ≤ θ ≤ } in the xy -planeas illustrated in Figure 3 and let Ω α = S α × [0 , M ] be a sloping beach domain. The water surfacexy α S α I I Figure 3.
The angular sector S α .is given by I S = I × [0 , M ] and the bottom of the beach is given by I N = I × [0 , M ] where I = { θ = 0 } and I = { θ = − α } . The sloping beach problem corresponds to finding a velocitypotential Φ : Ω α → R such that Φ is harmonic inside Ω α , satisfies Neumann boundary conditionson I N and the Steklov boundary condition ∂ ν Φ = σ Φ on I S . By separating variables, we get that Φ = ϕ ( x, y ) cos λ n z with λ n = nπM and ϕ satisfying(9) ∆ ϕ = λ n ϕ in S α ,∂ ν ϕ = 0 on I ,∂ ν ϕ = σϕ on I . We will create an approximate solution of (3) by using solutions from the sloping beach problem(9) coming from each corner of Σ . These solutions will need to meet smoothly and give rise to thesame eigenvalue. This gluing condition will then determine the possible quasi-eigenvalues.3.2. Edge wave solutions of the sloping beach problem.
Let < α ≤ π and n ∈ N .The edge wave solutions of the sloping beach problem (9) given by Ursell [21] are as follows. For ≤ m ≤ π α − , m ∈ Z , let ϕ nm ( x, y ) = e − λ n ( x cos α − y sin α ) + m (cid:88) j =1 A jm (cid:16) e − λ n ( x cos(2 j − α + y sin(2 j − α ) + (10) + e − λ n ( x cos(2 j +1) α − y sin(2 j +1) α ) (cid:17) where A jm = ( − j (cid:81) jr =1 tan( m − r +1) α tan( m + r ) α . One can check that ϕ nm solves (9) with(11) σ nm = λ n sin(2 m + 1) α. Note that if n = 0 , we get the constant solution and we can ignore it. In other words, there areno edge waves in the two-dimensional sloshing problem. We are particularly interested in the casewhere α = π q , in which case π α − = q − . In order to study these solutions, we need some estimateson ϕ nm and its derivatives. Lemma 3.1.
Let α = π q for an integer q ≥ . There exist positive constants C and c such that thefollowing estimates hold for all ( x, y ) in S α . For ≤ m < q − , (12) | ϕ nm ( x, y ) | ≤ Ce − cλ n x and (13) (cid:12)(cid:12) ∇ ( x,y ) ϕ nm ( x, y ) (cid:12)(cid:12) ≤ Cλ n e − cλ n x ; If q is odd, then for m = q − , (14) (cid:12)(cid:12)(cid:12) ϕ nm ( x, y ) − A mm e λ n y (cid:12)(cid:12)(cid:12) ≤ Ce − cλ n x and (15) (cid:12)(cid:12)(cid:12) ∇ ( x,y ) (cid:16) ϕ nm ( x, y ) − A mm e λ n y (cid:17)(cid:12)(cid:12)(cid:12) ≤ Cλ n e − cλ n x . Proof.
We will abuse notation slightly when using C and c throughout the proof, but they willalways denote positive constants depending only on the angle α .The first estimate (12) will follow from showing that for each exponential in (10), the sameestimate holds. Since y ≤ and ≤ (2 j + 1) α < π for j < π α , the estimate clearly holds for thefirst and third terms in (10). It remains to show that for all ≤ j < q − ,(16) x cos(2 j − α + y sin(2 j − α ≥ cx for some c > . The condition on j obviously only makes sense as long as q ≥ . We can rewrite (2 j − α = jπq − π q as π − ( q − j +1) π q to get x cos(2 j − α + y sin(2 j − α = x sin ( q − j + 1) π q + y cos ( q − j + 1) π q . Since y ∈ S α , we have − x tan α ≤ y ≤ and hence x cos(2 j − α + y sin(2 j − α ≥ x sin ( q − j + 1) π q − x cos ( q − j + 1) π q tan π q ≥ (cid:18) sin ( q − j + 1) π q − sin π q (cid:19) x where we used that cos ( q − j +1)2 q < cos π q since q − j ≥ . The constant before x in that lastexpression is strictly positive and therefore (16) holds.The estimate (13) follows from (12) since differentiating each term in (10) with respect to x or y introduces only a factor of at most λ n .Now if q is odd and m = q − , we have (2 m + 1) α = π and hence the last term in the sum defining ϕ nm is given by A mm e λ n y . By the previous calculations, all the other terms satisfy similar estimatesto (12) and (13). The sum of those terms is precisely ϕ nm ( x, y ) − A mm e λ n y , and hence both (14)and (15) hold. (cid:3) SYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM 11
Edge wave quasimodes.
We use the edge wave solutions of the sloping beach problem toconstruct solutions for the sloshing problem. To do so, we aim to glue together solutions comingfrom each corner. Notice that if a solution vanishes quickly outside its corresponding corner, wedon’t need to glue a solution coming from the other corner since it’ll simply correspond to the zerosolution near the other corner. However, if a solution does not vanish, then we have to be carefulsince there might not be a solution coming from the other corner for that eigenvalue.Let α = π q and β = π r . Denote by ϕ αnm (respectively ϕ βnm ) the edge wave solution of the slopingbeach problem coming from angle α with eigenvalue σ αnm = λ n sin(2 m + 1) α for ≤ m < q − .If q is even, every ϕ αnm ( x, y ) vanishes exponentially fast outside the corner α by Lemma 3.1,and therefore we can consider them as quasimodes individually. The same applies if r is even forthe solutions coming from angle β that are given in Σ by ϕ βn(cid:96) ( L − x, y ) with eigenvalue σ βn(cid:96) = λ n sin(2 (cid:96) + 1) β for ≤ (cid:96) < r − .If q is odd, then as above the solution ϕ αnm is a valid quasimode as long as m (cid:54) = q − . However,when m = q − , by Lemma 3.1, the solution ϕ αnm tends to A αmm on the surface y = 0 with acorresponding eigenvalue λ n . In order to get a valid quasimode, there should be a non-zero solutioncoming from the corner β with the same eigenvalue. This is only possible if r is also odd. In thatcase, we consider the quasimode(17) ψ n ( x, y ) = A β(cid:96)(cid:96) ϕ αnm ( x, y ) + A αmm ϕ βn(cid:96) ( L − x, y ) − A αmm A β(cid:96)(cid:96) e λ n where (cid:96) = r − . The last term is present so that we can control | ∂ ν ψ n | on W . We will use a similartrick for the surface wave quasimodes.In short, given n ∈ N , we constructed (cid:4) q (cid:5) and (cid:4) r (cid:5) quasimodes coming from the corners α and β respectively, as well as an additional quasimode if both q and r are odd. Remark 3.2.
Interestingly, our resulting edge wave quasimodes on the whole domain Ω oscillateonly along the edges of length M , but not those of length L . The computations of Section 2 confirmthat this phenomenon occurs when α = β = π . It should also hold for all the other triangularprisms and is motivated by the fact that the sloping beach problem has a single edge wave solutionwhen α = π given by e λ n y , which is constant along the sloshing edge I . Hence, one could expectthere to be solutions oscillating along an edge of length L if the wall adjacent to it met the sloshingsurface at an angle smaller than π .3.4. Surface wave solutions of the sloping beach problem.
Let us now construct surfacewave solutions of the sloping beach problem. To do so, we generalize the method used in [14]. Byrescaling in the z variable and by setting µ := λ n /σ , the problem (9) is equivalent to solving(18) ∆ ϕ = µ ϕ in S α ,∂ ν ϕ = 0 on I ,∂ ν ϕ = ϕ on I . However, recall that we are still solving to find the possible values of σ and although it doesn’tappear in the last formulation, it is actually hidden in µ .Let ξ = − π/q , and for a, b ∈ R , let g a,b denote the function g a,b ( x, y ) = e x cos( a )+ y sin( a ) e i √ − µ ( x cos( b )+ y sin( b )) . We define the linear operators A and B by ( A g a,b )( x, y ) := e x cos( − a + ξ )+ y sin( − a + ξ ) e i √ − µ ( x cos( − b + ξ )+ y sin( − b + ξ )) = g − a + ξ, − b + ξ ( x, y ) and ( B g a,b )( x, y ) := C a,b e x cos( a ) − y sin( a ) e i √ − µ ( x cos( b ) − y sin( b )) = C a,b g − a, − b ( x, y ) where C a,b = sin a + i (cid:112) − µ sin b − a + i (cid:112) − µ sin b + 1 . For an arbitrary function u on S α , we define its Steklov defect by SD( u ) := ( ∂ ν u − u ) | I . Note that
SD( u ) = 0 if and only if u satisfies the Steklov condition on I with eigenvalue . Bysimple calculations, one can show that these operators have the following useful properties. Proposition 3.3.
Let g be as above. We have (1) ( g − A g ) | I = 0 , (2) ∂ ν ( g + A g ) | I = 0 , (3) SD( g + B g ) = 0 . We will use these properties to construct a suitable function on S α . Let f ( x, y ) = e y e − i √ − µ x ,i.e. f is given by g π ,π . For ≤ m ≤ q − , we construct the functions f m = (cid:40) A f m − if m is odd , B f m − if m is even . Finally, we let v α = q − (cid:88) m =0 f m . The function v α is our main interest. In fact, it is a solution of (18)! Theorem 3.4.
The function v α as defined above satisfies ∆ v α = µ v α in S α , the Neumann conditionon I and SD( v α ) = 0 . In other words, it is a solution of (18) .Proof. First off, we can see that for any choice of a, b ∈ R , we have ∆ g a,b = (cid:104) µ + 2 i (cid:112) − µ cos( a − b ) (cid:105) g a,b . Since f = g π ,π , we have ∆ f = µ f . Both A and B act on g a,b by scaling and modifyingthe coefficients a and b , but keep the value of | a − b | unchanged. Then since f m is obtained byconsecutively applying A and B on f , we also have ∆ f m = µ f m for all m . By linearity, it thenfollows that ∆ v α = µ v α .For the Neumann condition, we see that we can write v α as v α = q − (cid:88) m =0 ( f m + f m +1 ) = q − (cid:88) m =0 ( f m + A f m ) and therefore, by Proposition 3.3, ∂ ν v α | I = q − (cid:88) m =0 ∂ ν ( f m + A f m ) | I = 0 . It remains to show that
SD( v α ) = 0 . We now write v α as v α = f + q − (cid:88) m =1 ( f m − + f m ) + f q − = f + q − (cid:88) m =1 ( f m − + B f m − ) + f q − SYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM 13 and therefore, by Proposition 3.3 and linearity of the Steklov defect,
SD( v α ) = SD( f ) + SD( f q − ) . Since f = e y e − i √ − µ x , we easily see that SD( f ) = 0 . Let us now show that SD( f q − ) = 0 . Forany choice of a and b , ( BA ) g a,b = C − a + ξ, − b + ξ g a − ξ,b − ξ . Hence, since f q − = A ( BA ) q − f = A ( BA ) q − g π ,π , we get(19) f q − = A q − (cid:89) j =1 C − π + jξ, − π + jξ g π − ( q − ξ,π − ( q − ξ = γ ( ξ ) g qξ − π ,qξ − π where γ ( ξ ) := q − (cid:89) j =1 C − π + jξ, − π + jξ . Since ξ = − π/q , we get f q − = γ ( ξ ) g − π , − π = γ ( ξ ) e y e i √ − µ x and thus SD( f q − ) = 0 . It followsthat SD( v α ) = 0 . (cid:3) In the previous proof, we started to compute f q − . Moving forward, we will need its exactexpression. Lemma 3.5.
The function f q − is given by γ ( ξ ) e y e i √ − µ x where γ ( ξ ) = ( − q − exp i q − (cid:88) j =1 arctan (cid:32) (cid:112) − µ sin jπq )cos jπq − (cid:33) . Proof.
The expression of f q − follows from (19). Moreover, γ ( ξ ) = q − (cid:89) j =1 sin( − π − jπq ) + i (cid:112) − µ sin( − π − jπq ) − − π − jπq ) + i (cid:112) − µ sin( − π − jπq ) + 1= q − (cid:89) j =1 − cos jπq + i (cid:112) − µ sin jπq − − cos jπq + i (cid:112) − µ sin jπq + 1= ( − q − q − (cid:89) j =1 cos jπq + i (cid:112) − µ sin jπq − jπq − i (cid:112) − µ sin jπq − where we have reordered the terms in the numerator by j (cid:55)→ q − j to get the last expression. Thedenominator is the complex conjugate of the numerator. Therefore, | γ ( ξ ) | = 1 and arg γ ( ξ ) = ( q − π + q − (cid:88) j =1 (cid:32) (cid:112) − µ sin jπq cos jπq − (cid:33) . The claim readily follows. (cid:3)
Lemma 3.6.
There exist positive constants C and c such that for all ( x, y ) ∈ S α , v α ( x, y ) = e y e − i √ − µ x + γ ( ξ ) e y e i √ − µ x + v d α ( x, y ) , with (20) (cid:12)(cid:12)(cid:12) v d α ( x, y ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∇ ( x,y ) v d α ( x, y ) (cid:12)(cid:12)(cid:12) ≤ Ce − cx . In particular, on the boundary I the solution v α ( x, y ) takes the form v α ( x ) = e − i √ − µ x + γ ( ξ ) e i √ − µ x + decaying exponentials . Proof.
As in the proof of Lemma 3.1, we will abuse notation throughout the proof when using C and c , but they again denote positive constants depending only on the angle α .The function v d α is given by (cid:80) q − m =1 f m . Therefore, it suffices to show that each of these f m satisfiesthe same estimate as (20). For each such f m , | f m | = F m e x cos( a )+ y sin( a ) for some constant F m > and a = ± ( π + j πq ) where j ∈ { , . . . , q − } , resulting from the successiveapplications of A and B . By periodicity, it is equivalent that a either takes values of the form π + jπq or π − jπq for ≤ j ≤ (cid:106) j (cid:107) . If a = π + jπq , then sin( a ) ≥ and cos( a ) < . Therefore, since y ≤ by definition of S α , we get | f m | ≤ F m e x cos( a ) = Ce − cx for C = F m and c = − cos( a ) . Now if a = π − jπq , then x cos( a ) + y sin( a ) = x cos (cid:18) π − jπq (cid:19) + y sin (cid:18) π − jπq (cid:19) = − x sin jπq − y cos jπq ≤ − x sin jπq + x cos jπq tan π q where we used in the last line that y ≥ − x tan α by definition of S α . Since cos jπq < cos π q , it followsthat | f m | ≤ Ce − cx for C = F m and c = sin jπq − sin π q > . Combining our estimates on f m for ≤ m ≤ q − , we can find positive constants C and c suchthat (cid:12)(cid:12) v d α ( x, y ) (cid:12)(cid:12) ≤ Ce − cx . Now by computing explicitly the derivitives of g a,b , one can show that | ∂ x g a,b | + | ∂ y g a,b | ≤ | g a,b | and hence given our previous estimates on the functions f m in v d α , we can find positive constants C and c such that (cid:12)(cid:12) ∇ ( x,y ) v d α ( x, y ) (cid:12)(cid:12) ≤ Ce − cx . Combining both estimates on v d α yields the result. (cid:3) Surface wave quasimodes.
We can now use the surface wave solutions of the sloping beachproblem to construct approximate solutions (quasimodes) for the sloshing problem on Σ . Let σ bea real scaling factor. We consider the functions v α ( σx ) and v β ( σ ( L − x )) corresponding to solutionsof the sloping beach problem starting off from the angles α and β respectively. Let v p α and v d α correspond to the principal part and decaying parts of v α on the boundary I (as in Lemma 3.6).In order for the sloping beach solutions to meet smoothly on S , we want their principal parts tomatch. Therefore, we look for σ such that(21) v p α ( σx ) = Qv p β ( σ ( L − x )) . SYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM 15 for some non-zero Q ∈ C . We call this the quantization condition. It fixes the values of σ and leadsto the quasimodes on Σ given by(22) g σ ( x, y ) = v α ( σx, σy ) + Qv d β ( σ ( L − x ) , σy ) = Qv β ( σ ( L − x ) , σy ) + v d α ( σx, σy ) . Notice that g σ satisfies ∆ g σ = µ σ g σ = λ n g σ in Σ and ∂ ν g σ = σg σ on S , but ∂ ν g σ (cid:54) = 0 on W andhence it is not exactly a solution of (3). However, we have ∂ ν v α = 0 on the side making the angle α with S , as well as ∂ ν v β = 0 on the side making the angle β . The error term in ∂ ν g σ on eachside of W therefore comes from the decaying part of the solution coming from the other side, whichvanishes exponentially by Lemma 3.6. Hence, the solution g σ is very close to being a solution of(3). 4. Counting of quasi-eigenvalues
Let N e and N s denote the counting functions for the edge wave and surface wave quasi-eigenvaluesrespectively. The total counting function for quasi-eigenvalues then becomes N e + N s .4.1. Counting the edge wave quasi-eigenvalues.
Recall that for ≤ m < q − the quasi-eigenvalue of the edge wave quasimode ϕ αnm ( x, y ) coming from the corner α is σ αnm = nπM sin(2 m +1) α .Therefore, the eigenvalue counting function for one such quasimode is given by { n ∈ N : σ αnm < σ } = M σπ sin(2 m + 1) α + O (1) . For ≤ (cid:96) < r − , we have a similar expression for the eigenvalue counting function of each edgewave quasimode ϕ βn(cid:96) ( L − x, y ) coming from the corner β .If q and r are both odd, we constructed another edge wave quasimode with eigenvalue nπM . Hence,if we let ν α,β := qr mod 2 , the total eigenvalue couting function for the edge wave quasimodes isgiven by N e ( σ ) = ν α,β M σπ + (cid:98) q − (cid:99) (cid:88) m =0 M σπ sin(2 m + 1) α + (cid:98) r − (cid:99) (cid:88) (cid:96) =0 M σπ sin(2 (cid:96) + 1) β + O (1) which is precisely the statement of Theorem 1.5.An interesting thing to note is that the expression for N e only depends on the angles and M , thelength of the side where the angles are on Ω . It does not depend on L . This makes sense since thesolutions mainly live along the side of length M by Lemma 3.1.4.2. Finding the surface wave quasi-eigenvalues.
Suppose that α = π q and β = π r . ByLemma 3.6, the principal part of v α ( σx ) is given by v p α ( σx ) = e − i √ − µ σx + γ ( ξ ) e i √ − µ σx where we can write γ ( ξ ) = ( − q − e iθ α for(23) θ α ( n, σ ) = q − (cid:88) j =1 arctan (cid:32) (cid:112) − µ sin jπq cos jπq − (cid:33) = − q − (cid:88) j =1 arctan (cid:113) − (cid:0) nπσM (cid:1) sin jπq − cos jπq . We have substituted µ = nπσM in the last equation. We have similar expressions for v β . Sincemultiplying v α and v β by constants still results in solutions of (18), we consider rather the functions V α and V β where V α ( x ) = (cid:40) e − iθ α v α ( x ) if q is odd, ie − iθ α v α ( x ) if q is even, with V β defined similarly. Notice that if q is odd, then the principal part of V α is given by V p α ( x ) = 2 cos( (cid:112) − µ x + θ α ) and if q is even, V p α ( x ) = 2 sin( (cid:112) − µ x + θ α ) . The quantization condition (21) then becomes V α ( σx ) = ± V β ( σ ( L − x )) which reduces to solving(24) (cid:114) − (cid:16) nπσM (cid:17) σL = − ( θ α + θ β ) + ( m − κ α,β ) π for m ∈ Z and κ α,β = (cid:40) if q and r have the same parity, otherwise.We can rewrite this equation as(25) σ = (cid:32) ( m − κ α,β − π ( θ α + θ β )) πL (cid:33) + (cid:16) nπM (cid:17) . It is important to keep in mind that θ α and θ β depend on σ and this is what makes the equationdifficult to solve. In the case where α = β = π , notice that equation (25) coincides with theequation (8) that we obtained from exact computation of the eigenfunctions. When κ α,β = 0 , thetrivial solution m = 0 and σ = nπM corresponds to the constant solution and we can ignore it. Wewish to restrict ourselves to positive values of m but since − ( θ α + θ β ) ≥ , we see that m can takenegative values in (24). However, there is only a finite number of such solutions. Lemma 4.1.
There is at most a finite number of pairs ( m, n ) with m ≤ ≤ n such that (24) admitsa nontrivial solution. Furthermore, for all m > and n ≥ , there exists a unique solution σ m,n of (24) .Proof. First, we show the case α = β = π q . For n ∈ N , consider the functions f n : [ nπM , ∞ ) → R defined by f n ( σ ) = 1 π (cid:32)(cid:114) − (cid:16) nπσM (cid:17) σL + 2 θ α ( n, σ ) (cid:33) . Notice that f n ( nπM ) = 0 and f n ( σ ) tends to infinity as σ → ∞ . Moreover, we can write(26) f (cid:48) n ( σ ) = 1 π (cid:113) − (cid:0) nπσM (cid:1) L − (cid:16) nπM (cid:17) q − (cid:88) j =1 α j σ [1 + α j (cid:16) − (cid:0) nπσM (cid:1) (cid:17) ] with α j = sin jπq − cos jπq > . When σ increases, the value of the sum strictly decreases and tends tozero. Hence, even if f (cid:48) n ( σ ) < for some values, it is eventually positive and tends to Lπ with thederivative vanishing at most once. When n gets sufficiently large, so does σ , and the derivative is SYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM 17 positive for all values of σ . In fact, when σ → nπM + , the expression in parentheses in (26) behaveslike L − σ q − (cid:88) j =1 α j which is positive for σ sufficiently large. Hence there exists n such that f (cid:48) n ( σ ) > for all σ > nπM and n > n .We see that σ is a solution of (24) corresponding to given integers m and n if and only if f n ( σ ) = m .From the previous calculations, there is only a finite number of f n which take negative values andthe set f − n (( −∞ , is bounded since f n tends to infinity as σ → ∞ . If f n takes negative values,it can then only take a finite number of negative integer values, and since its derivative vanishesexactly once, f n can be a given negative integer at most twice. Therefore, the set ∞ (cid:91) n =0 f − n ( Z < ) is finite and the first part of the lemma follows since we ignore the solutions with σ = nπM and m = 0 .The second part of the lemma follows from the fact that f (cid:48) n ( σ ) > whenever f n ( σ ) > and that f n tends to infinity.The proof with α (cid:54) = β is similar. Indeed, we only need to change one θ α by θ β + κ α,β in thedefinition of f n ( σ ) . It is straightforward to see that f (cid:48) n ( σ ) is eventually positive for all n sufficientlybig and since f n (cid:0) nπM (cid:1) = κ α,β , there is still a finite number of negative solutions. (cid:3) Counting the surface wave quasi-eigenvalues.
Now that we know how to find the surfacewave quasi-eigenvalues, we can count them in order to prove Theorem 1.6.We know from Lemma 4.1 that there is only a finite number of solutions corresponding to non-positive values of m . They contribute O (1) to the counting function and we can ignore them.Therefore, we restrict ourselves to solutions corresponding to m > and n ≥ . We also know thatfor each such pair ( m, n ) , there exists a unique solution of (24). We denote it by σ m,n . Let σ > and consider the set E σ = (cid:26) ( x, y ) ∈ R : (cid:16) xπσL (cid:17) + (cid:16) yπσM (cid:17) < (cid:27) . We have(27) { ( m, n ) ∈ E σ ∩ ( N × N ) } = LM π σ + L − M π σ + o ( σ ) where the error term o ( σ ) comes from known estimates on the Gauss circle problem (see [16] forexample). Suppose that ( m, n ) ∈ E σ and let d > be the horizontal distance between ( m, n ) andthe boundary ellipse of E σ , i.e. d = x n − m where x n is the positive solution to (cid:0) x n πσL (cid:1) + (cid:0) nπσM (cid:1) = 1 . From equation (25), we see that σ m,n < σ if and only if m + f ( n, σ m,n ) < x n or equivalently d > f ( n, σ m,n ) where f ( n, σ ) = − κ α,β − π ( θ α ( n, σ ) + θ β ( n, σ )) . Notice that f ( n, σ ) only depends on nσ and can hence be written as f ( nσ ) . We will use both notations.Therefore, counting the surface wave eigenvalues is equivalent (up to O (1) ) to counting the total number of integer points ( m, n ) ∈ E σ with m > and n ≥ to which we subtract the points suchthat d ≤ f ( n, σ m,n ) . Denote by N s − ( σ ) the number of such points, i.e. N s − ( σ ) = { ( m, n ) ∈ E σ ∩ ( N × N ) : d ≤ f ( n, σ m,n ) } . From equation (27), it then follows that N s ( σ ) = LM π σ + L − M π σ − N s − ( σ ) + o ( σ ) and therefore proving Theorem 1.6 is equivalent to proving the following. Theorem 4.2.
The counting function N s − ( σ ) satisfies N s − ( σ ) = M σπ (cid:90) f ( t ) d t + o ( σ ) . We start by giving an heuristic for this result. Let σ m,n be such that ( m, n ) ∈ E σ but σ m,n ≥ σ .We expect σ m,n to be relatively close to σ in a way that f ( n, σ m,n ) should be close to f ( n, σ ) . Forsimplicity of the argument, suppose that f ( n, σ m,n ) = f ( n, σ ) . The boundary of the ellipse E σ inthe first quadrant of the ( x, y ) plane can be given by the curve τ σ ( t ) = σLπ (cid:115) − (cid:18) tπσM (cid:19) , t . for t ∈ [0 , Mσπ ] . Let γ σ : [0 , σMπ ] be the curve γ σ ( t ) = τ σ ( t ) − ( f ( t, σ ) , . Then, the integer points in E σ in the region bounded by γ σ , τ σ and the x -axis are precisely thosesuch that d ≤ f ( n, σ m,n ) , i.e. those that contribute to N s − ( σ ) . It is then reasonable to expect thatthe area of this region should be a good approximation for the number of integer points within it.The area is given by (cid:90) Mσπ f (cid:18) tσ (cid:19) d t = M σπ (cid:90) f ( t ) d t. However, it could be that this approximation is not good at all since we took the area of a verythin strip which could miss all the integer points. For this estimate to be good, we need to showthat the integer points are well-behaved, in the sense that they are evenly or uniformly distributedacross this strip. To do so, we will rely on Weyl’s equidistribution theorem.In order to simplify the expressions, we now assume that L = M = π and α = β . However, theproofs will hold for all values. We will need the following two lemmas. Lemma 4.3.
For all m such that x n − q + 1 ≤ m ≤ x n and all ≤ n ≤ σ , the estimate f (cid:18) nσ m,n (cid:19) = f (cid:16) nσ (cid:17) + o (1) holds uniformly in m and n as σ → ∞ .Proof. Since ≤ f ( t ) ≤ q − from equation (23) for all t ∈ [0 , , and σ m,n = (cid:18) m + f (cid:18) nσ m,n (cid:19)(cid:19) + n it follows that ( x n − q + 1) + n ≤ σ m,n ≤ ( x n + q − + n SYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM 19 for m satisfying x n − q + 1 ≤ m ≤ x n . Expanding each side and using the fact that σ = x n + n yields σ − q − x n + ( q − ≤ σ m,n ≤ σ + 2( q − x n + ( q − . Since ± q − x n + ( q − = O ( σ ) , it follows that σ m,n = σ + O ( σ ) and hence σ m,n = σ + O (1) . Therefore, nσ m,n = nσ + O (1) = nσ + O (cid:16) nσ (cid:17) . Since ≤ n ≤ σ , we get that nσ m,n = nσ + O (cid:18) σ (cid:19) uniformly in n (and m ). Since f is uniformly continuous, it follows that, as σ → ∞ , f (cid:18) nσ m,n (cid:19) = f (cid:18) nσ + O (cid:18) σ (cid:19)(cid:19) = f (cid:16) nσ (cid:17) + o (1) . (cid:3) Lemma 4.4.
Fix K ∈ N and let h ∈ Z with h (cid:54) = 0 . Let e ( x ) = e πix . Then lim σ →∞ Kσ (cid:88) rσK ≤ n< ( r +1) σK e ( h (cid:112) σ − n ) = 0 for all ≤ r ≤ K − . To prove this lemma, we will need the following theorem from van der Corput [22] on boundingexponential sums.
Theorem 4.5 (van der Corput [22], 1922) . Let F : I → R be a C function on an interval I with λ ≤ | F (cid:48)(cid:48) ( x ) | ≤ αλ . Then (cid:88) n ∈ I e ( F ( n )) (cid:28) α | I | λ / + λ − / where the implied constant is absolute.Proof of Lemma 4.4. The following proof is inspired by a proof provided to us by Zeev Rudnick.We apply Theorem 4.5 with I = (cid:104) rσK , ( r +1) σK (cid:17) and F ( x ) = h √ σ − x . We have F (cid:48)(cid:48) ( x ) = − hσ ( σ − x ) / . Since σ − x ≤ σ , we have | h | σ ≤ (cid:12)(cid:12) F (cid:48)(cid:48) ( x ) (cid:12)(cid:12) . On the other hand, since r ≤ K − , we have σ − x > σ − ( r +1) σK ≥ σK and hence (cid:12)(cid:12) F (cid:48)(cid:48) ( x ) (cid:12)(cid:12) = | h | σ (( σ − x )( σ + x )) / ≤ | h | σ ( σ K ) / = K / | h | σ . Fixing h and applying Theorem 4.5 with λ = | h | σ and α = K / yields (cid:88) n ∈ I e ( F ( n )) (cid:28) h K / σK √ σ + √ σ = √ σ ( √ K + 1) It follows that Kσ (cid:88) n ∈ I e ( h (cid:112) σ − n ) (cid:28) h K / + K √ σ which tends to as σ → ∞ . (cid:3) Denote by d n ( σ ) the distance between x n (the positive solution of σ = x n + n ) and the closestinteger point ( m, n ) satisfying m + n < σ . This distance is precisely the fractional part of √ σ − n . From Weyl’s equidistribution theorem, Lemma 4.4 is equivalent to the following lemmawhich will enable us to prove Theorem 4.2. Lemma 4.6.
Fix K ∈ N . Then, for any interval [ α, β ] ⊂ [0 , and for all ≤ r ≤ K − , lim σ →∞ Kσ (cid:26) n ∈ (cid:20) rσK , ( r + 1) σK (cid:19) : d n ( σ ) ∈ [ α, β ] (cid:27) = β − α. Proof of Theorem 4.2.
We wish to estimate N s − ( σ ) = (cid:88) ( m,n ) ∈ E σ { x n − m ≤ f ( n, σ m,n ) } since σ m,n < σ if and only if x n − m > f ( n, σ m,n ) . Since f is bounded by q − , we have that { x n − m ≤ f ( n, σ m,n ) } = 0 for all m such that m < x n − q + 1 . Hence, N s − ( σ ) = (cid:98) σ (cid:99) (cid:88) n =0 (cid:98) x n (cid:99) (cid:88) m = (cid:100) x n − q +1 (cid:101) { x n − m ≤ f ( n, σ m,n ) } . From Lemma 4.3, for the values of n and m present in the sum, we can find a function h ( σ ) whichgoes to zero as σ → ∞ such that(28) { x n − m ≤ f ( n, σ ) − h ( σ ) } ≤ { x n − m ≤ f ( n, σ m,n ) } ≤ { x n − m ≤ f ( n, σ ) + h ( σ ) } . This motivates us to rather estimate the quantity S ( σ ) := (cid:98) σ (cid:99) (cid:88) n =0 (cid:98) x n (cid:99) (cid:88) m = (cid:100) x n − q +1 (cid:101) { x n − m ≤ f ( n, σ ) } . Writing m = (cid:98) x n (cid:99) − r , this is equivalent to (cid:98) σ (cid:99) (cid:88) n =0 (cid:98) x n (cid:99)−(cid:100) x n − q +1 (cid:101) (cid:88) r =0 { x n − (cid:98) x n (cid:99) ≤ f ( n, σ ) − r } . Since ≤ x n − (cid:98) x n (cid:99) < , we see that { x n − (cid:98) x n (cid:99) ≤ f ( n, σ ) − r } = if r ≤ (cid:98) f ( n, σ ) (cid:99) − { x n − (cid:98) x n (cid:99) ≤ f ( n, σ ) − r } if r = (cid:98) f ( n, σ ) (cid:99) if r > (cid:98) f ( n, σ ) (cid:99) + 1 . SYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM 21
Since f is strictly decreasing, it takes integer values at most q − times. With a small error, wecan therefore change the last condition to r ≥ (cid:98) f ( n, σ ) (cid:99) + 1 . We then get S ( σ ) = (cid:98) σ (cid:99) (cid:88) n =0 (cid:98) f ( n, σ ) (cid:99) + { x n − (cid:98) x n (cid:99) ≤ f ( n, σ ) − (cid:98) f ( n, σ ) (cid:99)} + O (1) We now consider S ( σ ) σ . We claim that lim σ →∞ S ( σ ) σ = (cid:90) f ( t ) d t. Rewriting f ( n, σ ) as f (cid:0) nσ (cid:1) , the first term of S ( σ ) yields lim σ →∞ σ (cid:98) σ (cid:99) (cid:88) n =0 (cid:106) f (cid:16) nσ (cid:17)(cid:107) = (cid:90) (cid:98) f ( t ) (cid:99) d t. Setting g (cid:0) nσ (cid:1) = f (cid:0) nσ (cid:1) − (cid:4) f (cid:0) nσ (cid:1)(cid:5) and noticing that x n − (cid:98) x n (cid:99) = d n ( σ ) , it remains to estimate lim σ →∞ σ (cid:98) σ (cid:99) (cid:88) n =0 (cid:110) d n ( σ ) ≤ g (cid:16) nσ (cid:17)(cid:111) . Let ε > and let K ∈ N be such that K < ε and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K K − (cid:88) r =0 g ( x r ) − (cid:90) g ( t ) d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε for all choices of x r ∈ (cid:2) rK , r +1 K (cid:3) . Such a K exists since g is piecewise continuous. Dividing [0 , σ ] into K subintervals, we get that σ (cid:98) σ (cid:99) (cid:88) n =0 (cid:110) d n ( σ ) ≤ g (cid:16) nσ (cid:17)(cid:111) ≤ K K − (cid:88) r =0 Kσ n ∈ (cid:20) rσK , ( r + 1) σK (cid:19) : d n ( σ ) ≤ sup x ∈ [ rK , r +1 K ] g ( x ) . The reverse inequality holds with the supremum replaced with the infimum. When r = K − , wecan use the trivial bound Kσ n ∈ (cid:20) rσK , ( r + 1) σK (cid:19) : d n ( σ ) ≤ sup x ∈ [ rK , r +1 K ] g ( x ) ≤ . However, when ≤ r ≤ K − , we can use Lemma 4.6. Together, this yields(29) lim σ →∞ σ (cid:98) σ (cid:99) (cid:88) n =0 (cid:110) d n ( σ ) ≤ g (cid:16) nσ (cid:17)(cid:111) ≤ K K − (cid:88) r =0 sup x ∈ [ rK , r +1 K ] g ( x ) + 1 K < (cid:90) g ( t ) d t + ε. Proceeding similarly with the reversed inequality, it follows that for all ε > , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim σ →∞ σ (cid:98) σ (cid:99) (cid:88) n =0 (cid:110) d n ( σ ) ≤ g (cid:16) nσ (cid:17)(cid:111) − (cid:90) g ( t ) d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε and therefore lim σ →∞ S ( σ ) σ = (cid:90) (cid:98) f ( t ) (cid:99) d t + (cid:90) f ( t ) − (cid:98) f ( t ) (cid:99) d t = (cid:90) f ( t ) d t. Finally, we see that if we were to change f ( n, σ ) for f ( n, σ ) ± h ( σ ) with h ( σ ) going to as σ → ∞ in the definition of S ( σ ) , the result would still hold since (29) holds from the fact that forall y ∈ [0 , , lim σ →∞ Kσ (cid:26) n ∈ (cid:20) rσK , ( r + 1) σK (cid:19) : d n ( σ ) ≤ y + o (1) (cid:27) = y. From (28), it then follows that lim σ →∞ N s − ( σ ) σ = (cid:90) f ( t ) d t so that N s − ( σ ) = σ (cid:90) f ( t ) d t + o ( σ ) . (cid:3) Quasimode analysis and numerical evidence
The results we have presented are only approximate solutions of problem (2). However, we willshow that there is an actual eigenvalue of the problem near every quasi-eigenvalue and our numericalexperiments seem to agree with both our conjectures.5.1.
Analysis of the quasi-eigenvalues.
For n ∈ N , let { ˜ σ ( n ) j } j ∈ N denote the set of quasi-eigenvalues (coming from both our edge waves and surface waves solutions) indexed in ascendingorder for which the quasimodes solve ∆ ϕ = λ n ϕ in Σ , and let { σ ( n ) k } k ∈ N denote the set of realeigenvalues (sloshing eigenvalues) of problem (3). The following lemma is analogous to Lemma . in [14]. Lemma 5.1.
There exist positive constants C and c such that for every n ∈ N and j ∈ N , thereexists k ∈ N such that (30) (cid:12)(cid:12)(cid:12) ˜ σ ( n ) j − σ ( n ) k (cid:12)(cid:12)(cid:12) ≤ Ce − c ˜ σ ( n ) j . In order to prove it, we need a preliminary result on our quasimodes. We denote by ϕ σ aquasimode with quasi-eigenvalue σ . Proposition 5.2.
There exist positive constants C and c such that for any quasimode ϕ σ , | ∂ ν ϕ σ | ≤ Ce − cσ for all ( x, y ) ∈ W .Proof. Let us denote by W α and W β the segments of W making angles α and β with S respectively.We will again abuse notation when using C and c and we will use the fact that C σe − c σx ≤ C e − c σ whenever x is bounded from below by a positive number.Firstly, if ϕ σ is an edge wave quasimode of the form ϕ αnm with m (cid:54) = q − , then ∂ ν ϕ σ = 0 on W α .Moreover, by Lemma 3.1, since σ ≥ λ n sin π q , we can find C, c > such that | ∂ ν ϕ σ | ≤ Ce − cσ on W β . The same reasoning applies if ϕ σ is an edge wave quasimode of the form ϕ βn(cid:96) with (cid:96) (cid:54) = r − .Secondly, if ϕ σ is the edge wave quasimode given by ψ n as in (17), then on W β , we have | ∂ ν ψ n | = (cid:12)(cid:12)(cid:12) ∂ ν ( A α(cid:96)(cid:96) ϕ αnm ( x, y ) − A αmm A β(cid:96)(cid:96) e λ n y ) (cid:12)(cid:12)(cid:12) SYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM 23 since ( ∂ ν ϕ βn(cid:96) ) | W β = 0 . Applying the estimate (15) from Lemma 3.1 to that last expression yields | ∂ ν ψ n | ≤ Ce − cσ on W β . A similar reasoning yields the same estimate on W α , and therefore on all W .Finally, if ϕ σ = g σ is a surface wave quasimode given by equation (22), then by using the secondexpression for g σ , we see that we have on W β | ∂ ν g σ | = (cid:12)(cid:12)(cid:12) ∂ ν ( v d α ( σx, σy )) (cid:12)(cid:12)(cid:12) since ( ∂ ν v β ) | W β = 0 . The estimate on the gradient of v d α in Lemma 3.6 gives us our desired boundon W β . By using the first expression for g σ , we can do the same reasoning on W α , showing that | ∂ ν g σ | ≤ Ce − cσ everywhere on W .In all our calculations, both C and c depend solely on the angles α and β . The claim thenfollows. (cid:3) Proof of Lemma 5.1.
We will follow the argument laid out in Section . of [14] and slightly adaptit to our case. We refer to [14] for further details of the argument.Given one of our quasimodes ϕ σ satisfying ∆ ϕ σ = λ n ϕ σ in Σ and ∂ ν ϕ σ = σϕ σ on S , consider afunction η σ that is solution of(31) ∆ η σ = λ n η σ in Σ ,∂ ν η σ = ∂ ν ϕ σ on W ,∂ ν η σ = − (cid:0)(cid:82) W ∂ ν ϕ σ (cid:1) ψ on S , where ψ ∈ C ∞ ( S ) is a fixed function supported away from the the corners α and β with (cid:82) S ψ = 1 .The function η σ is the result of the Neumann-to-Dirichlet map ND − λ n : L ( ∂ Σ) → L ( ∂ Σ) whenapplied to the function h σ = (cid:40) ∂ ν ϕ σ on W , − (cid:0)(cid:82) W ∂ ν ϕ σ (cid:1) ψ on S . When n = 0 , as mentioned in [14], such a solution η σ exists up to a constant and is thereforeunique if we demand that (cid:82) ∂ Σ η σ = 0 . Moreover, when acting on functions with mean-value on S , ND is bounded. Now if n > , the operator ND − λ n is well-defined since − λ n < is not aNeumann eigenvalue of − ∆ on Σ and it is a self-adjoint compact operator on L ( ∂ Σ) [3]. Moreover,the operators ND − λ n are uniformly bounded on L ( ∂ Σ) since their eigenvalues decrease when n increases. This is due to the fact that ND − λ is the inverse of the Dirichlet-to-Neumann map DN − λ whose eigenvalues are positive and strictly increasing for λ in the interval ( ε, ∞ ) , see [2] or [6]. Itfollows from Proposition 5.2 that(32) (cid:107) η σ (cid:107) L ( S ) ≤ (cid:13)(cid:13) ND − λ n h σ (cid:13)(cid:13) L ( ∂ Σ) ≤ C (cid:107) h σ (cid:107) L ( ∂ Σ) ≤ Ce − cσ where the constants do not depend on n nor σ .The function v σ := ϕ σ − η σ satisfies ∆ v σ = λ n v σ and its normal derivative vanishes on W . Let DN − λ n now denote the Dirichlet-to-Neumann map that takes f ∈ L ( S ) and maps it to ( ∂ ν ˜ f ) | S where ∆ ˜ f = λ n ˜ f in Σ , ∂ ν ˜ f = 0 on W , and ˜ f = f on S . Then, by construction, we have DN − λ n ( v σ | S ) = ( ∂ ν v σ ) | S . Since ∂ ν ϕ σ = σϕ σ on S , for every quasi-eigenvalue σ we have(33) (cid:13)(cid:13) DN − λ n ( v σ | S ) − σv σ (cid:13)(cid:13) L ( S ) = (cid:107) ∂ ν η σ − ση σ (cid:107) L ( S ) ≤ Ce − cσ where the last inequality follows from (32) and Proposition 5.2. By rescaling, suppose now that (cid:107) v σ (cid:107) L (Γ S ) = 1 and let ( φ ( n ) k ) k ≥ be a complete set of orthonormaleigenfunctions of DN − λ n with eigenvalues σ ( n ) k . Then, we can find coefficients a k = ( v σ , φ k ) suchthat (cid:80) ∞ k =0 a k = 1 and v σ = ∞ (cid:88) k =0 a k φ k . It follows from (33) that (cid:13)(cid:13) DN − λ n ( v σ | S ) − σv σ (cid:13)(cid:13) L ( S ) = ∞ (cid:88) k =0 a k ( σ ( n ) k − σ ) ≤ Ce − cσ and since (cid:80) ∞ k =0 a k = 1 , there must be a k such that ( σ ( n ) k − σ ) ≤ Ce − cσ and therefore(34) (cid:12)(cid:12)(cid:12) σ − σ ( n ) k (cid:12)(cid:12)(cid:12) ≤ Ce − cσ . Plugging σ = ˜ σ ( n ) j into (34) yields (30). (cid:3) We now have all the tools to prove Theorem 1.8.
Proof of Theorem 1.8.
We start by showing N ( σ ) ≥ N s ( σ ) + o ( σ ) . In order to get this estimate, weneed to show that every surface wave quasi-eigenvalue is sufficiently isolated in order for every actualeigenvalue given by Lemma 5.1 to be distinct. Denote the set of surface wave quasi-eigenvalues thatsolve (3) for a given n by { σ ( n ) j } j ∈ N . First of all, given n (cid:54) = n (cid:48) , we know that the real eigenvaluescorresponding to σ ( n ) j and σ ( n (cid:48) ) j (cid:48) are distinct eigenvalues of problem (2) for all j, j (cid:48) ∈ N , since thecorresponding eigenfunctions solve the equation ∆ u = λu in Σ for different values of λ . By distinct,we do not necessarily mean that the eigenvalues are not equal, but rather that they correspond todifferent linearly independent eigenfunctions.Recall that σ > nπM is a quasi-eigenvalue of a surface wave ϕ σ satisfying ∆ ϕ σ = λ n σ if and only if f n ( σ ) = 1 π (cid:32)(cid:114) − (cid:16) nπσM (cid:17) σL + θ α ( n, σ ) + θ β ( n, σ ) (cid:33) + κ α,β is an integer (see Lemma 4.1 and its proof). Moreover, there exists n ∈ N such that for all n ≥ n the function f n : (cid:2) nπM , ∞ (cid:1) → R is always positive and its derivative strictly decreases and tends to Lπ . Therefore, for n ≥ n , the eigenvalues σ ( n ) j satisfy f n ( σ ( n ) j ) = j. By convexity of f n , it follows that(35) (cid:12)(cid:12)(cid:12) σ ( n ) j +1 − σ ( n ) j (cid:12)(cid:12)(cid:12) ≥ σ ( n )1 − nπM where f n ( σ ( n )1 ) = 1 . Since θ α and θ β are both negative, we have f n ( x ) ≤ h n ( x ) := 1 π (cid:114) − (cid:16) nπxM (cid:17) xL + κ α,β for all x ≥ nπM . Letting x = (cid:115)(cid:18) π (1 − κ α,β ) L (cid:19) + (cid:16) nπM (cid:17) SYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM 25 we see that h n ( x ) = 1 . Since h n is strictly increasing, it follows that f n ( x ) < for all x < x andtherefore σ ( n )1 ≥ x . Consequently, σ ( n )1 − nπM ≥ x − nπM ≥ Cn where C can be chosen independently of n . From (35), we get (cid:12)(cid:12)(cid:12) σ ( n ) j +1 − σ ( n ) j (cid:12)(cid:12)(cid:12) ≥ Cn .
Hence, using that σ ( n ) j > nπM , Lemma 5.1 guarantees that given n sufficiently large the real eigenvaluenext to σ ( n ) j is distinct for each j ∈ N .Now suppose that n isn’t large enough for the previous approach to apply. We know that f (cid:48) n tends to Lπ and so there exists j ( n )0 ∈ R such that σ ( n ) j = πL ( j − j ( n )0 ) + o n ( j ) . Therefore, there exists a constant C n such that for all j sufficiently large (cid:12)(cid:12)(cid:12) σ ( n ) j +1 − σ ( n ) j (cid:12)(cid:12)(cid:12) ≥ C n . Since σ ( n ) j ≥ Cj , Lemma 5.1 then guarantees that if j and j (cid:48) are sufficiently large, the sloshingeigenvalues next to σ ( n ) j and σ ( n ) j (cid:48) are distinct as long as j (cid:54) = j (cid:48) .In short, all the sloshing eigenvalues σ ( n ) k given by Lemma 5.1 close to the surface wave quasi-eigenvalues σ ( n ) j are distinct as long as either n or j is sufficiently large. Thus, only a finite numberof such sloshing eigenvalues can be identical. Denote that number by P . Then, we have N s ( σ − Ce − cσ ) − P ≤ N ( σ ) for all σ ≥ . Our knowledge of N s ( σ ) guarantees that N s ( σ − Ce − cσ ) = N s ( σ ) + o ( σ ) , which yields N ( σ ) ≥ N s ( σ ) + o ( σ ) .Let us now consider the edge wave quasimodes. As in the case of the surface wave quasi-eigenvalues, the sloshing eigenvalues given by Lemma 5.1 for different values of n have to be distinctsince the underlying eigenfunctions solve different equations inside Σ .We consider first the quasimodes ϕ αnm and ϕ βn(cid:96) for n ∈ N , ≤ m < q − and ≤ (cid:96) < r − , withquasi-eigenvalues given by σ αnm = λ n sin(2 m + 1) α and σ βn(cid:96) = λ n sin(2 (cid:96) + 1) β. If there are values of m and (cid:96) such that (2 m + 1) r = (2 (cid:96) + 1) q , then some quasi-eigenvalues σ αnm and σ βn(cid:96) have multiplicity and we will deal with them afterwards. Suppose for now that there areno such values of m and (cid:96) . Then, there exists δ > such that, given n , every edge wave quasi-eigenvalue is spaced by δ and at distance at least δ from nπM . Lemma 5.1 then guarantees that,except for maybe a finite number of them, all the real eigenvalues associated to those edge wavequasi-eigenvalues are distinct, and distinct from the ones we recovered close to the surface wavequasi-eigenvalues.If q and r are both odd, we also have to consider the quasimodes ψ n with eigenvalue λ n = nπM .Since σ ( n )1 − nπM ≥ Cn and each other edge wave quasi-eigenvalue σ αnm or σ βnm is at a distance at least δ from nπM , it follows from Lemma 5.1 that, except for maybe a finite number of them, all the realeigenvalues close to a quasi-eigenvalue λ n are distinct from the ones we found previously.Suppose now that there exist m < q − and (cid:96) < r − such that (2 m + 1) r = (2 (cid:96) + 1) q . In otherwords, suppose that there are edge wave quasi-eigenvalues with multiplicity 2 since σ αnm = σ βn(cid:96) for all n ∈ N . Let us show that the multiplicity guarantees the presence of two distinct sloshingeigenvalues. Fix n ∈ N and let ϕ α , ϕ β and σ denote respectively ϕ αnm , ϕ βn(cid:96) and σ αnm . Now let v α = ϕ α − η α where η α is the solution of (31) for ϕ σ = ϕ α . Rescaling if need be, suppose further that v α has unitnorm in L ( S ) . Then, by (33) and Theorem 4.1 in [14], we can find a function w α such that • w α is a linear combination of eigenfunctions of DN − λ n with eigenvalues in the interval [ σ − √ Ce − cσ/ , σ + √ Ce − cσ/ ] ; • (cid:107) w α (cid:107) L ( S ) = 1 ; • (cid:107) v α − w α (cid:107) L ( S ) ≤ √ Ce − cσ/ (1 + o σ (1)) .Here, C and c are the same constants as in Lemma 5.1. Divide the boundary S into two parts S α = [0 , L/ × { } and S β = ( L/ , L ] × { } . Then, we have (cid:107) w α (cid:107) L ( S β ) ≤ (cid:107) ϕ α (cid:107) L ( S β ) + (cid:107) η α (cid:107) L ( S ) + (cid:107) w α − v α (cid:107) L ( S ) . By Lemma 3.1, equation (33) and the definition of w α , each of the terms on the right-hand side ofthe last equation vanish exponentially fast as σ (and therefore n ) goes to infinity. It follows that (cid:107) w α (cid:107) L ( S β ) goes to as n → ∞ . We can repeat all of the previous construction for the angle β toget a function w β with the same properties as w α but with respect to v β = ϕ β − η β . By the samearguments as above, (cid:107) w β (cid:107) L ( S α ) goes to as n → ∞ and therefore (cid:107) w β (cid:107) L ( S β ) = (cid:107) w β (cid:107) L ( S ) − (cid:107) w β (cid:107) L ( S α ) goes to since (cid:107) w β (cid:107) L ( S ) = 1 . Both w α and w β have unit norm in L ( S ) , but (cid:107) w α (cid:107) L ( S β ) → while (cid:107) w β (cid:107) L ( S β ) → . Thus, for n sufficiently large, the two functions must be linearly independent.It follows that there are at least two eigenfunctions of DN − λ n with eigenvalues in the interval [ σ − √ Ce − cσ/ , σ + √ Ce − cσ/ ] . For n sufficiently large, those eigenvalues must be distinct from allthe previous sloshing eigenvalues that we found previously. Therefore, there are indeed 2 distinctsloshing eigenvalues close to each edge wave quasi-eigenvalue of multiplicity 2 that is sufficientlylarge.Since the sloshing eigenvalues from Lemma 5.1 that are close to the edge wave and surface wavequasimodes are distinct (except for maybe a finite number of them), we can combine them usingthe same trick we used for comparing N s and N . This yields N ( σ ) ≥ N s ( σ ) + N e ( σ ) + o ( σ ) as claimed. (cid:3) Discussion on quasimodes.
We have shown that the counting function of our quasimodesbounds the real eigenvalue counting function from below, but in order to prove Conjecture 1.9,we also need to prove that it bounds it from above. This should require showing that our quasi-eigenvalues approximate all the sloshing eigenvalues, which should be much more difficult to proveand require new ideas. In dimension , it turns out that the quasimodes solve a Sturm-Liouvilleequation on the sloshing part of the boundary. This fact was used in [14] to show that theirquasimodes formed a complete set, and hence approximated every eigenfunction. Their methodcould work in our case, but we were unable to find an analogous Sturm-Liouville equation solvedby our quasimodes. Furthermore, the presence of edge waves makes it even more complicated. SYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM 27
We only considered the cases where the angles α and β were of the form π q . Note that ourconstruction of the edge wave quasimodes is valid for any angle smaller than π . However, we usedthe fact that the angles were of the form π q to construct explicitly the surface wave solutions of thesloping beach problem that we used in our quasimodes. Indeed, if we were to repeat the steps inSection 3.4 for an arbitrary angle which is a rational multiple of π , the iterations of the operators A and B would lead to solutions that blow up at infinity and an analogous version of Lemma 3.6would not hold. There might be a way to remedy this, but we were unable to do so. Moreover,we are unsure how the counting function behaves for arbitrary angles. In two dimensions, solutionsdue to Peters [17] allow to create quasimodes for arbitrary angles. Using the ideas of Peters in [18],it should be possible to find similar solutions in three dimensions, which could lead to finding anexpression of N ( σ ) for arbitrary angles.5.3. Numerical evidence supporting Conjectures 1.9 and 1.11.
We now present numericalevidence to support both our conjectures. Let Σ be the triangle of angles α and β with sidelength L resulting from the separation of variable on Ω (as in Figure 1). We used FreeFem++ to solveproblem (3) using the finite element method. It is a 2-d problem and hence much faster to solvethan its 3-d counterpart of solving directly problem (2) on all Ω .For simplicity, we take L = M = π . We start by computing N ( σ ) up to σ = 50 for all thecombinations of α and β in the set { π , π , π } . In order to do so, we compute the first eigenvaluescorresponding to λ n = n for sufficiently many n ’s. We order and denote those eigenvalues by σ k ( n ) . Note that from a theorem by Friedlander [6], the eigenvalue σ k ( n ) gets larger as n increases.Therefore, we only need to compute these eigenvalues until σ ( n ) > and we can reduce thenumber of computed eigenvalues at each step in order to speed up the computations.Consider the function S ( σ ) := 1 σ (cid:18) N ( σ ) − LM π σ (cid:19) = 1 σ (cid:16) N ( σ ) − π σ (cid:17) . Then, Conjecture 1.9 is equivalent to showing lim σ →∞ S ( σ ) = N s (1) + N e (1) − π where here N s and N e are the expressions from Theorems 1.6 and 1.5 without the error terms.The plots in Figure 4 show our estimated value of S ( σ ) for ≤ σ ≤ , as well as the value of N e (1) + N s (1) − π to which it should converge when σ tends to infinity.When computing the eigenvalues numerically, we found that our quasi-eigenvalues matched themquite accurately. We have an exact expression for the edge wave quasi-eigenvalues from equation(11) and we can compute the surface wave quasi-eigenvalues by solving equation (24) for differentvalues of m (without forgetting that m can take negatives values if n is small). We did so usingthe function FindRoot in Mathematica. Tables 2, 3, 4 show the first quasi-eigenvalues computedwith Mathematica as well as the first sloshing eigenvalues computed with FreeFEM++ for differentvalues of α and β . As we conjectured, our quasi-eigenvalues seem to be asymptotically completesince they match the sloshing eigenvalues starting from a certain index. We have shifted the tablesto highlight their matching. σ ( σ ) α = π , β = π σ ( σ ) α = π , β = π σ ( σ ) α = π , β = π σ ( σ ) α = π , β = π σ ( σ ) α = π , β = π σ ( σ ) α = π , β = π Figure 4.
Value of S ( σ ) compared to its conjectured limit indicated by the hori-zontal line. SYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 2.
The first 125 quasi-eigenvalues (on the left) and sloshing eigenvalues (onthe right) for α = π , β = π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 3.
The first 125 quasi-eigenvalues (on the left) and sloshing eigenvalues (onthe right) for α = π , β = π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 4.
The first 250 quasi-eigenvalues (on the left) and sloshing eigenvalues (onthe right) for α = π , β = π . SYMPTOTICS OF SLOSHING EIGENVALUES FOR A TRIANGULAR PRISM 31
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