2021 UNC Greensboro PDE Conference, Electronic Journal of Differential Equations, Conference 26 (2022), pp. 45–58. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ON THE L2-ORTHOGONALITY OF STEKLOV EIGENFUNCTIONS MANKI CHO, MAURICIO A. RIVAS Abstract. This article analyzes the interior L2-orthogonality of the Steklov eigenfunctions on rectangles Ω1α. It is shown that most Steklov eigenfunctions are, indeed, pairwise orthogonal in L2(Ω1α), and pairs that are not orthogonal are nearly orthogonal. Explicit formulae for exact inner products in L2(Ω1α) of the eigenfunctions are found, and to elucidate the intricate formulae obtained, accompanying numerics are provided. Then envelopes that bound the calcu- lated inner products are constructed that simplify the convoluted formulae. This leads to a straightforward description of the nearly orthogonal Steklov eigenfunctions. A consequence of the calculations is a tabulation of the mean value of Steklov eigenfunctions over Ω1α. 1. Introduction This article describes the exact, or near, orthogonality in L2(Ω1α) of the sequence of Steklov eigenfunctions in the case Ω1α is a rectangle in R2. This complements the well-known result of L2-orthogonality of the sequence of Dirichlet eigenfunctions, as well as of the Neumann and Robin eigensystems, on more general domains. The question on L2-orthogonality follows from [6], where Auchmuty and the second author analyzed the tensor product of pairs of these four systems. Classes of Steklov eigenfunctions have been used, for instance, in the construction of bases of trace spaces of functions on the boundary (Auchmuty [1] and Kloucek et al. [14]), on the analysis of dewetting of thin films (Auchmuty and Klouček [5]), on the spectral representation of divergence-free vector fields (Auchmuty and Simpkins [7]), and on harmonic boundary value problems (as done by the first author in [4, 8, 9, 10, 11]). To the best of the authors knowledge, a complete description of the orthogonality in L2(Ω), on more general regions Ω in RN including rectangles in R2, of Steklov eigenfunctions has not been made because results and applications often employ orthogonality in L2(∂Ω) or (special) orthogonality in other Sobolev- Hilbert spaces. After introducing notation in §2, and making precise in §3 the orthogonality issue studied here, collected in §4 is the explicit formulae for the harmonic Steklov eigendata on a rectangle Ω1α that is reprised from [3]. The main L2-orthogonality results of this paper are given in §5. 2020 Mathematics Subject Classification. 35P05, 31A20, 35J05. Key words and phrases. Harmonic functions; Steklov eigenfunctions; Laplacian eigenfunctions. ©2022 This work is licensed under a CC BY 4.0 license. Published August 25, 2022. 45 46 M. CHO, M. A. RIVAS EJDE-2022/CONF/26 First, the exact L2-norms on Ω1α and ∂Ω1α are recorded. Then explicit formu- lae for the inner product in L2(Ω1α) between Steklov eigenfunctions are given, and numerical work is shown to interpret the elaborate formulae. It is calcualted that the majority of the Steklov eigenfunctions are pairwise orthogonal in L2(Ω1α), and it is seen from the plots that those that are not orthogonal are nearly orthogonal at high frequencies. Calculation of the inner product of the constant Steklov eigen- function and other Steklov eigenfunctions is re-interpreted as a calculation of the mean value of the Steklov eigenfunctions. To provide straightforward formulae that indicate near L2-orthogonality of Steklov eigenfunctions, envelopes are constructed to bound the intricate formulae previously obtained. These envelope formulae are further simplified and this leads to the relation (5.16) that succinctly describes the near orthogonality in L2(Ω1α) of the sequence of Steklov eigenfunctions. The findings in the present paper may be generalized to the case Ω1α is a cuboid in N -dimensions; see Girouard et al. [12] where the Steklov spectrum is carefully analyzed for cuboids. We expect that similar results hold, but that the formulae would be intense. Our work is an analysis of special functions. 2. Assumptions and notation The analysis in this work will be over a retangle Ω1α := (−1, 1)× (−α, α) in R2, where α is a fixed constant in (0, 1] that is called the aspect ratio of the rectangle. Due to scaling and rotation properties of the Steklov problem, the analysis on Ω1α accounts for the Steklov analysis on any rectangle of R2. Denote by dσ the 1- dimensional Hausdorff measure, or arclength, so that the unit outward normal ν(z) is defined for σ a.e. z ∈ ∂Ω1α. All functions in this work will take value in [−∞,∞]. Let Lp(Ω1α) and Lp(∂Ω1α) with 1 ≤ p ≤ ∞, be the usual Lebesgue spaces with p-norm denoted by ‖u‖p,Ω1α or ‖u‖p,∂Ω1α respectively. When p = 2 these are real Hilbert spaces with inner products defined by 〈u, v〉2,Ω1α := ∫ Ω1α uv dx dy and 〈u, v〉2,∂Ω1α := ∫ ∂Ω uvdσ. Denote by H1(Ω1α) the usual real Sobolev space of functions on Ω1α that is a real Hilbert space under the standard H1-inner product [u, v]1,2,Ω1α = ∫ Ω1α [u · v +∇u · ∇v] dx dy (2.1) where ∇u is the gradient of the function u; the associated norm is denoted by ‖u‖1,2,Ω1α . The trace γu of a continuous function u on Ω1α to the boundary ∂Ω1α is its restriction to ∂Ω1α. The boundary trace map on H1(Ω1α) is the linear extension of the map γ restricting Lipschitz continuous functions on Ω1α to ∂Ω1α. The region Ω1α is said to satisfy a compact trace theorem provided that the trace mapping γ : H1(Ω1α)→ L2(∂Ω1α, dσ) is compact. One inequality that implies the compact trace theorem for bounded regions in RN with Lipschitz boundaries has been proved in [13, Theorem 1.5.1.10]. Instead of (2.1), one can use the ∂-inner product defined by [u, v]∂ := ∫ Ω1α ∇u · ∇v dx dy + 1 |∂Ω1α| ∫ ∂Ω1α uv dσ. (2.2) EJDE-2018/CONF/26 L2-ORTHOGONALITY OF STEKLOV EIGENFUNCTIONS 47 Here, |∂Ω1α| = 4(1 + α) is the length of the perimeter of the rectangle, and dσ is integration with respect to arclength. The norm corresponding to [·, ·]∂ is denoted by ‖u‖∂ . From Corollary 6.2 of [2], this norm is equivalent to the standard norm of H1(Ω1α). A function u ∈ H1(Ω1α) is said to be harmonic on Ω1α if it satisfies∫ Ω1α ∇u · ∇v dx dy = 0 for all v ∈ C1 c (Ω1α) (2.3) where C1 c (Ω1α) is the set of all C1-functions on Ω1α with compact support in Ω1α. Denote by H(Ω1α) the space of all such harmonic functions on Ω1α. The usual Sobolev space H1 0 (Ω1α) is the closure of C1 c (Ω1α) in the H1(Ω1α)-norm, and it is easy to see that H(Ω1α) is ∂-orthogonal to H1 0 (Ω1α) so that H1(Ω1α) may be expressed as H1(Ω1α) = H1 0 (Ω1α)⊕∂ H(Ω1α) (2.4) where ⊕∂ represents a ∂-orthogonal decomposition as described in §5 of [2]. 3. Steklov eigenfunctions and the L2-orthogonality question This article is about the harmonic Steklov eigenfunctions on Ω1α, which are non-zero functions s = s(x, y) in H1(Ω1α) satisfying, for some σ ∈ R, the identity∫ Ω1α ∇s · ∇v dx dy = σ |∂Ω1α| ∫ ∂Ω1α sv dσ for all v ∈ H1(Ω1α). (3.1) Equation (3.1) is the weak form of the boundary value problem ∆s = 0 in Ω1α and ∂u ∂n = σ |∂Ω1α| s on ∂Ω1α. In a quite general bounded region Ω of RN that includes rectangles, Auchmuty in [2] obtains a countable infinite sequence of Steklov eigenfunctions and proves, among other properties, that this sequence is orthogonal in H(Ω) with respect to the ∂-inner product, and that the corresponding sequence of traces is orthogonal in L2(∂Ω,dσ). Determining the orthogonality in L2(Ω1α) of the sequence of Steklov eigenfunc- tions on the rectangle Ω1α is what this paper investigates and provides various results. 4. Steklov eigendata on rectangles Ω1α of R2 This section recapitulates the explicit Steklov spectral data for rectangles Ω1α that is contained in Auchmuty-Cho [3]. There the eigendata on Ω1α is organized into four classes according to symmetry. Here the Steklov eigenfunctions are denoted by u instead of s to indicate that they are unnormalized. Class I Steklov eigenfunctions u = u(x, y) are even in x and in y. The first such function is given by u1,0(x, y) := 1, which corresponds to the zero eigenvalue σ1,0 := 0. Then there is a dichotomy for all other functions and values in this class given by u1i(x, y) := coshβix cosβiy corresponding to σ1i = βi tanhβi, i ∈ N, u1j(x, y) := cosβjx coshβjy corresponding to σ1j = βj tanhαβj , j ∈ N, 48 M. CHO, M. A. RIVAS EJDE-2022/CONF/26 where βi and βj are the ascending, strictly positive zeros, respectively, of tanαβ + tanhβ = 0 and tanβ + tanhαβ = 0. (4.1) These are called the determining equations in β for Class I Steklov eigendata. Steklov eigenfunctions u = u(x, y) in Class II are odd in x and in y. When α = 1, the first such function is given by u2,0(x, y) = xy, which corresponds to the eigenvalue σ2,0 := 1. All other eigendata in this class splits as u2i(x, y) := sinhβix sinβiy corresponding to σ2i = βi cothβi, i ∈ N, u2j(x, y) := sinβjx sinhβjy corresponding to σ2j = βj cothαβj , j ∈ N, where βi and βj , in this case, are the ascending, strictly positive zeros, respectively, of tanαβ − tanhβ = 0 and tanβ − tanhαβ = 0. (4.2) Now Class III functions are even in x and odd in y, and the class is separated as u3i(x, y) := coshβix sinβiy corresponding to σ3i = βi tanβi, i ∈ N, u3j(x, y) := cosβjx sinhβjy corresponding to σ3j = βj tanhαβj , j ∈ N, according to the respective determining equations tanαβ − cothβ = 0 and tanβ + cothαβ = 0. (4.3) Finally, Class IV functions are odd in x and even in y, and the the two subclasses are u4i(x, y) := sinhβix cosβiy corresponding to σ4i = βi cothβi, i ∈ N, u4j(x, y) := sinβjx coshβjy corresponding to σ4j = βj cothαβj , j ∈ N, according to the respective determining equations tanαβ + cothβ = 0 and tanβ − cothαβ = 0. (4.4) 5. Exact or near orthogonality of Steklov eigenfunctions in L2(Ω1α) This section analytically treats the main question of orthogonality in L2(Ω1α) of the harmonic Steklov eigenfunctions catalogued in §4. 5.1. Interior and boundary L2-norms of Steklov eigenfunctions on Ω1α. To calculate explicitly the orthogonality in L2(Ω1α) between Steklov eigenfunctions, their L2-norms on Ω1α and on ∂Ω1α were found and are listed in Table 1. Here, the indices i, j for βi, βj are suppresed in the formulae to elucidate the form of these norms, the u`i, u`j are the unnormalized Steklov eigenfunctions of §4, and the following functions have been used sinc θ := sin θ θ and sinhc θ := sinh θ θ (5.1) EJDE-2018/CONF/26 L2-ORTHOGONALITY OF STEKLOV EIGENFUNCTIONS 49 Table 1. L2-norms of (unnormalized) Steklov eigenfunctions u. u ‖u‖22,Ω1α ‖u‖22,∂Ω1α u1,0 4α 4(1 + α) u1i (1 + sinhc 2β)(α+ α sinc 2αβ) 2 cos2(αβ)[1 + sinhc 2β] + 2α cosh2(β)[1 + sinc 2αβ] u1j (1 + sinc 2β)(α+ α sinhc 2αβ) 2 cosh2(αβ)[1 + sinc 2β] + 2α cos2(β)[1 + sinhc 2αβ] u2,0 4/9 8/3 u2i (−1 + sinhc 2β)(α− α sinc 2αβ) 2 sin2(αβ)[−1 + sinhc 2β] + 2α sinh2(β)[1− sinc 2αβ] u2j (1− sinc 2β)(−α+ α sinhc 2αβ) 2 sinh2(αβ)[1− sinc 2β] + 2α sin2(β)[−1 + sinhc 2αβ] u3i (1 + sinhc 2β)(α− α sinc 2αβ) 2 sin2(αβ)[1 + sinhc 2β] + 2α cosh2(β)[1− sinc 2αβ] u3j (1 + sinc 2β)(−α+ α sinhc 2αβ) 2 sinh2(αβ)[1 + sinc 2β] + 2α cos2(β)[−1 + sinhc 2αβ] u4i (−1 + sinhc 2β)(α+ α sinc 2αβ) 2 cos2(αβ)[−1 + sinhc 2β] + 2α sinh2(β)[1 + sinc 2αβ] u4j (1− sinc 2β)(α+ α sinhc 2αβ) 2 cosh2(αβ)[1− sinc 2β] + 2α sin2(β)[1 + sinhc 2αβ] 5.2. Mean-Value of Steklov eigenfunctions on Ω1α. The inner products in L2(Ω1α) of the first Steklov eigenfunction u1,0 ≡ 1 with the other Steklov eigen- functions uli, ulj , where l = 1, 2, 3, 4 and i, j ∈ N, lead to the calculations 〈ũ1,0, ũ〉2,Ω1α =  2 sinhc βi sincαβi√ (1+sinhc 2βi)(1+sinc 2αβi) if ũ = ũ1i 2 sinc βj sinhcαβj√ (1+sinc 2βj)(1+sinhc 2αβj) if ũ = ũ1j 0 if u = u2,0, u2i, u2j , u3i, u3j , u4i, u4j (5.2) where ũ indicates that u is normalized with respect to the standard norm of L2(Ω1α). The graphs of βi 7→ 〈ũ1,0, ũ1i〉2,Ω1α and βj 7→ 〈ũ1,0, ũ1j〉2,Ω1α are in Figure 1, where discrete points on the graphs are at the roots βi, βj that determine u1i, u1j , respectively. As seen in Figure 1(a), the L2(Ω1α)-angle between u1,0 and u1i, which are both Class I Steklov eigenfunctions, rapidly approaches 90◦ as i → ∞. From Figure 1(b), the same is true of the angle between u1,0 and u1j . The third case in (5.2) shows that u1,0 is orthogonal in L2(Ω1α) to Class II, III, and IV Steklov eigenfunctions. This result can be interpreted as a result on the mean value of each L2-normalized Steklov eigenfunction over the region Ω1α since 〈ũ1,0, ũ〉2,Ω1α = 1 4α ∫ Ω1α ũ dx (5.3) and 4α is the area of the rectangle Ω1α. In this language, the Class II, III, IV Steklov eigenfunctions have mean value zero on Ω1α, and the mean value on Ω1α of Class I Steklov eigenfunctions is nearly zero for high frequency u1i, u1j ; by high frequency is meant that i, j are large values. 5.3. L2-orthogonality of Steklov eigenfunctions on Ω1α. The inner product in L2(Ω1α) of a Class I Steklov eigenfunction u1i, with i fixed, and another Steklov eigenfunction, where the prime in u′1i indicates a second Steklov eigenfunction of the form u1i, is evaluated and leads to 50 M. CHO, M. A. RIVAS EJDE-2022/CONF/26 10 20 30 40 50 60 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (a) L2-inner product between ũ1,0 and ũ1i; this equals the mean value of ũ1i over Ω1α 10 20 30 40 50 60 -0.2 0 0.2 0.4 0.6 0.8 (b) L2-inner product between ũ1,0 and ũ1j ; this equals the mean value of ũ1j over Ω1α Figure 1. Near L2-orthogonality between ũ1,0 and Steklov eigen- functions from Class I on Ω1α shown for α = 0.5, 0.8, 1.0. 〈ũ1i, ũ〉2,Ω1α =  [sinhc(βi+β ′ i)+sinhc(βi−β′i)]·[sinc(α(βi+β ′ i))+sinc(α(βi−β′i))]√ (1+sinhc 2βi)(1+sinc 2αβi)(1+sinhc 2β′i)(1+sinc 2αβ′i) if ũ = ũ′1i 4[βi sinh βi cos βj+βj cosh βi sin βj ][βi sin(αβi) cosh(αβj)+βj cos(αβi) sinh(αβj)] α·(β2 i+β2 j )2· √ (1+sinhc 2βi)(1+sinc 2αβi)(1+sinc 2βj)(1+sinhc 2αβj) if ũ = ũ1j 0 if ũ = ũ2,0, ũ2i, ũ2j , ũ3i, ũ3j , ũ4i, ũ4j . (5.4) To facilitate the orthogonality discussion, the symbols ⊥ and f will be used for the phrases is orthogonal to and is nearly orthogonal to, respectively. With this notation, the calculation in (5.4) shows that u1i ⊥ u`i and u1i ⊥ u`j in L2(Ω1α) for ` = 2, 3, 4, and that u1i f u′1i and u1i f u1j in L2(Ω1α) for high frequency u′1i and u1j ; see Figure 2. These and the next computations for 〈ũ`i , ũ〉2,Ω1α are for fixed i; analogous formulae hold when i is replaced by j. The L2(Ω1α)-inner product of u2i, a fixed Class II, Type 1 Steklov eigenfunction, with another Steklov eigenfunction is found and yields EJDE-2018/CONF/26 L2-ORTHOGONALITY OF STEKLOV EIGENFUNCTIONS 51 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 i |〈ũ 1 1 , ũ 1 i〉 | α = 0.5 α = 0.8 α = 1.0 (a) L2-inner product between ũ11 and ũ1i for 2 ≤ i ≤ 30 on Ω1α 5 10 15 20 25 30 0 5 · 10−2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 j |〈ũ 1 1 , ũ 1 j 〉| α = 0.5 α = 0.8 α = 1.0 (b) L2-inner product between ũ11 and ũ1j for 1 ≤ j ≤ 30 on Ω1α Figure 2. Near L2-orthogonality of Class I Steklov eigenfunction on Ω1α shown for α = 0.5, 0.8, 1.0. 〈ũ2i, ũ〉2,Ω1α =  [sinhc(βi+β ′ i)−sinhc(βi−β′i)][sinc(α(βi−β′i))−sinc(α(βi+β ′ i))]√ (−1+sinhc 2βi)(1−sinc 2αβi)(−1+sinhc 2β′i)(1−sinc 2αβ′i) if ũ = ũ′2i 4[βi cosh βi sin βj−βj sinh βi cos βj ][βj sin(αβi) cosh(αβj)−βi cos(αβi) sinh(αβj)] α·(β2 i+β2 j )2· √ (−1+sinhc 2βi)(1−sinc 2αβi)(1−sinc 2βj)(−1+sinhc 2αβj) if ũ = ũ2j 0 if ũ = ũ3i, ũ3j , ũ4i, ũ4j (5.5) 52 M. CHO, M. A. RIVAS EJDE-2022/CONF/26 This shows u2i ⊥ u`i and u2i ⊥ u`j in L2(Ω1α) for ` = 3, 4, and that u2i f u′2i and u2i f u2j in L2(Ω1α) for high frequency u′2i and u2j ; see Figure 3. 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 5 · 10−2 0.1 0.15 0.2 0.25 i |〈ũ 2 1 , ũ 2 i〉 | α = 0.5 α = 0.8 α = 1.0 (a) L2-inner product between ũ21 and ũ2i for 2 ≤ i ≤ 30 on Ω1α 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 j |〈ũ 2 1 , ũ 2 j 〉| α = 0.5 α = 0.8 α = 1.0 (b) L2-inner product between ũ21 and ũ2j for 1 ≤ j ≤ 30 on Ω1α Figure 3. Near L2-orthogonality of Class II Steklov eigenfunc- tions on Ω1α shown for α = 0.5, 0.8, 1.0. EJDE-2018/CONF/26 L2-ORTHOGONALITY OF STEKLOV EIGENFUNCTIONS 53 The inner product in L2(Ω1α) of u3i, a fixed Class III, Type 1 Steklov eigenfunc- tion, with another Steklov eigenfunction is computed and leads to 〈ũ3i, ũ〉2,Ω1α =  [sinhc(βi−β′i)+sinhc(βi+β ′ i)][sinc(α(βi−β′i))−sinc(α(βi+β ′ i))]√ (1+sinhc 2βi)(1−sinc 2αβi)(1+sinhc 2β′i)(1−sinc 2αβ′i) if ũ = ũ′3i 4[βi sinh βi cos βj+βj cosh βi sin βj ][βj sin(αβi) cosh(αβj)−βi cos(αβi) sinh(αβj)] α·(β2 i+β2 j )2· √ (1+sinhc 2βi)(1−sinc 2αβi)(1+sinc 2βj)(−1+sinhc 2αβj) if ũ = ũ3j 0 if ũ = ũ4i, ũ4j (5.6) Thus, u3i ⊥ u4i and u3i ⊥ u4j in L2(Ω1α), and u3i f u′3i and u3i f u3j in L2(Ω1α) for high frequency u′3i and u3j ; see Figure 4. Lastly, the inner product in L2(Ω1α) of u4i, a fixed Class IV, Type 1 Steklov eigenfunction, with another Class IV Steklov eigenfunction is evaluated and gives 〈ũ4i, ũ〉2,Ω1α =  [sinhc(βi+β ′ i)−sinhc(βi−β′i)]·[sinc(α(βi−β′i))+sinc(α(βi+β ′ i))]√ (−1+sinhc 2βi)(1+sinc 2αβi)(−1+sinhc 2β′i)(1+sinc 2αβ′i) if ũ = ũ′4i 4[βi cosh βi sin βj−βj sinh βi cos βj ][βj cos(αβi) sinh(αβj)+βi sin(αβi) cosh(αβj)] α·(β2 i+β2 j )2· √ (−1+sinhc 2βi)(1+sinc 2αβi)(1−sinc 2βj)(1+sinhc 2αβj) if ũ = ũ4j (5.7) This says u4i f u′4i and u4i f u4j in L2(Ω1α) for high frequency u′4i and u4j ; see Figure 5. 5.4. Envelopes for the L2-orthogonality. Although exact formulae for inner products have been found, admittedly the expressions are formidable. The next result provides easier formulae for envelopes, or bounds, on these inner products, with the envelopes given in terms of the aspect ratio α of the rectangle Ω1α and the roots βi, βj that determine the Steklov data. Theorem 5.1. Let ũ1,0 and ũ`,i, with ` = 1, 2, 3, 4, be the Steklov eigenfunctions normalized in L2(Ω1α) as described above. (1) When the root βi that determines ũ1i satisfies βi > max{ e −2βi 4 , 1 2α}, we have |〈ũ1,0, ũ1i〉2,Ω1α | ≤ 2 √ 2 βi √ α(2αβi − 1) (5.8) (2) When the root βj that determines ũ1j satisfies βj > max{ e −2αβj 4α , 1 2} , we have |〈ũ1,0, ũ1j〉2,Ω1α | ≤ 2 √ 2 βj √ α(2βj − 1) (5.9) (3) When the roots βi, β ′ i that determine ũ1i, ũ ′ 1i satisfy β′i > 2 α + βi and βi > max{e −2βi 4 , 1 2α } and β′i > max {e−2β′i 4 , 1 2α } 54 M. CHO, M. A. RIVAS EJDE-2022/CONF/26 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 5 · 10−2 0.1 0.15 0.2 0.25 0.3 0.35 i |〈ũ 3 1 , ũ 3 i〉 | α = 0.5 α = 0.8 α = 1.0 (a) L2-inner product between ũ31 and ũ3i for 2 ≤ i ≤ 30 on Ω1α 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 j |〈ũ 3 1 , ũ 3 j 〉| α = 0.5 α = 0.8 α = 1.0 (b) L2-inner product between ũ31 and ũ3j for 1 ≤ j ≤ 30 on Ω1α Figure 4. Near L2-orthogonality of Class III Steklov eigenfunc- tions on Ω1α shown for α = 0.5, 0.8, 1.0. with α < 1, we have |〈ũ1i, ũ ′ 1i〉2,Ω1α | ≤ 16βiβ ′ i (βi − β′i)2 √ (2αβi − 1)(2αβ′i − 1) (5.10) EJDE-2018/CONF/26 L2-ORTHOGONALITY OF STEKLOV EIGENFUNCTIONS 55 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 2 · 10−2 4 · 10−2 6 · 10−2 8 · 10−2 0.1 0.12 0.14 0.16 0.18 0.2 0.22 i |〈ũ 4 1 , ũ 4 i〉 | α = 0.5 α = 0.8 α = 1.0 (a) L2-inner product between ũ41 and ũ4i for 2 ≤ i ≤ 30 on Ω1α 5 10 15 20 25 30 0 5 · 10−2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 j |〈ũ 4 1 , ũ 4 j 〉| α = 0.5 α = 0.8 α = 1.0 (b) L2-inner product between ũ41 and ũ4j for 1 ≤ j ≤ 30 on Ω1α Figure 5. Near L2-orthogonality of Class IV Steklov eigenfunc- tions on Ω1α shown for α = 0.5, 0.8, 1.0. (4) When the roots βi, βj that determine ũ1i, ũ1j satisfy βi > max {e−2βi 4 , 1 2α } and βj > max {e−2αβj 4α , 1 2 } , 56 M. CHO, M. A. RIVAS EJDE-2022/CONF/26 we have |〈ũ1i, ũ1j〉2,Ω1α | ≤ 32(βi + βj) 2βiβj coshβi coshαβj eβi+αβj (β2 i + β2 j )2 √ (2αβi − 1)(2βj − 1) (5.11) Proof. Rewriting the formula (5.2) for normalized ũ1,0 and ũ1i from Class I, we obtain 〈ũ1,0, ũ1i〉2,Ω1α = 4 sinhβi sinαβi βi √ α √ (2βi + sinh 2βi)(2αβi + sin 2αβi) ≤ 2 √ 2 βi √ α √ 2αβi − 1 using sinαβi ≤ 1 and sinhβi ≤ ( eβi 2 ) to obtain the majorizing numerator, and using the relation βi > max{ e −2βi 4 , 1 2α} to obtain the smaller denominator at the end. Thus, the first assertion holds. An analogous majorization gives the envelope for 〈ũ1,0, ũ1j〉2,Ω1α . Using that sine and cosine are bounded above by one, in the formula (5.4) the numerator of 〈ũ1i, ũ1j〉2,Ω1α is majorized by 4[βi sinhβi + βj coshβi][βi cosh(αβj) + βj sinh(αβj)]. For βi, βj > 0, the relations sinhβi < coshβi and sinhαβj < coshαβj hold, which implies the numerator for 〈ũ1i, ũ1j〉2,Ω1α is majorized by 4[βi+βj ] 2 coshβi coshαβj . The denominator of 〈ũ1i, ũ1j〉2,Ω1α can be rewritten as (β2 i + β2 j )2 4βiβj √ (2βi + sinh 2βi)(2αβi + sin 2αβi)(2βj + sin 2βj)(2αβj + sinh 2αβj). When the roots βi, βj satisfy the prescribed inequalities, the factors under the square root are made smaller so that the denominator of 〈ũ1i, ũ1j〉2,Ω1α is minorized by (β2 i + β2 j )2eβi+αβj 8βiβj √ (2αβi − 1)(2βj − 1) These numerator and denominator results give the envelope for |〈ũ1i, ũ1j〉2,Ω1α | in the fourth assertion. The third assertion that gives the envelope for |〈ũ1i, ũ ′ 1i〉2,Ω1α | is a bit more tricky. From sinhc(βi + β′i) ≤ 1 2e βi+β ′ i βi + β′i ≤ 1 2e βi+β ′ i β′i − βi , which holds since β′i > βi, and from sinhc(βi − β′i) = sinhc(β′i − βi) ≤ 1 2e β′i−βi β′i − βi ≤ 1 2e β′i+βi β′i − βi it follows that sinhc(βi + β′i) + sinhc(βi − β′i) ≤ eβi+β ′ i β′i − βi . For the second factor in the numerator of |〈ũ1i, ũ ′ 1i〉2,Ω1α |, use sinc θ ≤ 1 θ for θ > 2, to obtain | sinc(α(βi − β′i))| = | sinc(α(β′i − βi))| ≤ 1 α(β′i − βi) , | sinc(α(βi + β′i))| ≤ 1 α(β′i + βi) ≤ 1 α(β′i − βi) EJDE-2018/CONF/26 L2-ORTHOGONALITY OF STEKLOV EIGENFUNCTIONS 57 Thus | sinc(α(βi − β′i))|+ | sinc(α(βi + β′i))| ≤ 2 α(β′i − βi) . The denominator for |〈ũ1i, ũ ′ 1i〉2,Ω1α | can be rewritten as 1 4αβiβ′i · √ (2βi + sinh 2βi)(2αβi + sin 2αβi)(2β′i + sinh 2β′i)(2αβ ′ i + sin 2αβ′i). When βi, β ′ i satisfy the prescribed inequalities, this denominator is minorized by eβi+β ′ i 8αβiβ′i √ (2αβi − 1)(2αβ′i − 1) Combining these numerator and denominator bounds and simplifying gives the third assertion. � Note that this theorem on envelopes provides exact inequalities for the inner products of Steklov eigenfunctions in Class I, and thus the envelope formulae are still somewhat intricate. However, a consequence of these bounding curves is the following succinct asymptotic estimates on the inner products. Corollary 5.2. Let ũ1,0 and ũ`,i, with ` = 1, 2, 3, 4, be the Steklov eigenfunctions normalized in L2(Ω1α) as described above. (1) When the root βi that determines ũ1,i satisfies βi > max{ e −2βi 4 , 1 2α}, we have |〈ũ1,0, ũ1i〉2,Ω1α | / 2 αβ 3/2 i . (5.12) (2) When the root βj that determines ũ1,j satisfies βj > max { e−2αβj 4α , 1 2 } , we have |〈ũ1,0, ũ1j〉2,Ω1α | / 2 √ αβ 3/2 j . (5.13) (3) When the roots βi, β ′ i that determine ũ1i, ũ ′ 1i satisfy β′i > 2 α + βi and βi > max {e−2βi 4 , 1 2α } and β′i > max {e−2β′i 4 , 1 2α } with α < 1, we have |〈ũ1i, ũ ′ 1i〉2,Ω1α | / 8 √ βiβ′i (βi − β′i)2 · α . (5.14) (4) When the roots βi, βj that determine ũ1i, ũ1j satisfy βi > max {e−2βi 4 , 1 2α } and βj > max {e−2αβj 4α , 1 2 } , we have |〈ũ1i, ũ1j〉2,Ω1α | / 4(βi + βj) 2 √ βiβj (β2 i + β2 j )2 √ α . (5.15) Going a step further, note that for high frequency Steklov eigenfunctions ũ1`, each of the four cases presented in the corollary simplify to the form |〈ũ1i, ũ1`〉2,Ω1α | / C · β−3/2 ` (5.16) for some constant C that depends on the aspect ratio α of Ω1α and the root βi that determines the first factor ũ1i. The relation (5.16) quantifies the statement 58 M. CHO, M. A. RIVAS EJDE-2022/CONF/26 that Class I Steklov eigenfunctions are nearly (pairwise) orthogonal in L2(Ω1α), and from the above results, Class I is in fact exactly orthogonal in L2(Ω1α) to all other classes. Acknowledgements. The authors would like to thank the reviewers for their thoughtful comments and efforts towards improving our manuscript. References [1] G. Auchmuty; Spectral characterization of the trace spaces Hs(∂Ω), SIAM J. Math. Anal., 38 (2006), 894–907. [2] G. Auchmuty; Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numer. Funct. Anal. 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Manki Cho Department of Mathematics and Statistics, University of Houston - Clear Lake, 2700 Bay Area Blvd, Houston, TX 77058, USA Email address: cho@uhcl.edu Mauricio A. Rivas Department of Mathematics and Statistics, North Carolina A&T State University, 1601 East Market Street, Greensboro, NC 27411, USA Email address: marivas@ncat.edu 1. Introduction 2. Assumptions and notation 3. Steklov eigenfunctions and the L2-orthogonality question 4. Steklov eigendata on rectangles 1 of R2 5. Exact or near orthogonality of Steklov eigenfunctions in L2(1) 5.1. Interior and boundary L2-norms of Steklov eigenfunctions on 1 5.2. Mean-Value of Steklov eigenfunctions on 1 5.3. L2-orthogonality of Steklov eigenfunctions on 1 5.4. Envelopes for the L2-orthogonality Acknowledgements References