Closed Form Inverses of the Weakly Singular and Hypersingular Operators on Disks

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(1)Integr. Equ. Oper. Theory (2018) 90:4 https://doi.org/10.1007/s00020-018-2425-y Published online February 28, 2018 c Springer International Publishing AG, part of Springer Nature 2018. Integral Equations and Operator Theory. Closed-Form Inverses of the Weakly Singular and Hypersingular Operators on Disks R. Hiptmair, C. Jerez-Hanckes and C. Urzúa-Torres Abstract. We establish new explicit expressions and variational forms of boundary integral operators that provide the exact inverses of the weakly singular and hypersingular operators for −Δ on flat disks. We derive closed-form formulas for their singular kernels. We also show that the inverse of the weakly singular operator can be obtained by composing surface curl operators and the inverse of the hypersingular operator. Mathematics Subject Classification. Primary 45P05, 31A10, 65R20; Secondary 65N38. Keywords. Boundary integral operators, Screens, Exact inverses, Projected spherical harmonics.. 1. Introduction We study the weakly singular and hypersingular Boundary Integral Operators (BIOs) arising when recasting screen problems in R3 into Boundary Integral Equations (BIEs). In particular, we consider the following singular BIEs on the flat disk Da of radius a > 0: 1 σ(x) (V σ)(y) := dDa (x) = g(y), (1.1) 4π Da x − y 1 1 ∂2 (W u)(y) := − u(x) dDa (x) = μ(y), (1.2) 4π Da ∂nx ∂ny x − y for y ∈ Da and functions σ, g, u, μ in suitable trace Sobolev spaces that will be made explicit later on. Here · stands for the standard Euclidean norm, ∂/∂n for the normal derivative in a direction perpendicular to the disk Da , and − designates finite-part integrals in distributional sense [1, Ch. 5]. Equations The work of the second author was partially funded by Chile CORFO Engineering 2030 program through Grant OPEN-UC 201603, Conicyt Anillo ACT1417 and Fondecyt Regular 1171491. The work of the third author was supported by ETHIRA Grant ETH-04 13-2..

(2) 4. Page 2 of 14. R. Hiptmair et al.. IEOT. (1.1) and (1.2) are connected with exterior Dirichlet and Neumann problems for the Laplacian −Δ in R3 \ Da [2, Sect. 3.5.3], respectively. Usually (1.1) is referred to as weakly singular BIE, while (1.2) is called hypersingular. In (1.1) and (1.2) we may replace Da with a connected orientable Lipschitz manifold with boundary of co-dimension one, a so-called screen Γ ⊂ R3 . A comprehensive theory of the arising BIOs in the framework of Sobolev spaces on screens is available [2–5]. 1.1. Motivation: Calderón Preconditioning Dual mesh-based Calderón Preconditioning is a recently invented technique addressing the ill-conditioning of matrices generated by the Galerkin discretization of first-kind BIEs by means of piecewise polynomial Boundary Element (BE) functions on fine surface meshes. For details we refer to the seminal paper [6] and to [7,8]. Tersely speaking, the method hinges on the availability of “preconditioning” BIOs that are (i) amenable to BE Galerkin discretization, that is, they may feature at most Cauchy-singular kernels that are easily evaluated, and (ii) spectrally equivalent to the inverses of the targeted BIOs. 1.2. Closed Versus Open Surfaces In (1.1) and (1.2), Da may also be replaced with the boundary ∂Ω of a bounded Lipschitz domain Ω ⊂ R3 . In this case the remarkable Calderón identities will hold [2, Prop. 3.6.4]. They tell us that V can serve as a preconditioning operator for W and viceversa, because both W V and V W provide bounded operators of trace spaces into themselves. The complementary mapping properties of V and W on ∂Ω are also reflected by the fact that Dirichlet and Neumann trace spaces, H 1/2 (∂Ω) and H −1/2 (∂Ω) enjoy L2 -duality. The situation on screens (open surfaces) is fundamentally different. Then the boundary integral operators V and W spawn continuous mappings −1/2 (Γ) → H 1/2 (Γ) and W : H 1/2 (Γ) → H −1/2 (Γ).1 Recall that the V : H ±1/2 norms on H (Γ) are strictly stronger than those on H ±1/2 (Γ). This clearly rules out Calderón identities in their standard form and preconditioning BIOs for V and W on Da are no longer straightforward. Of course, in the literature many different solution formulas for (1.1) and (1.2) have been proposed, some in the form of integrals [10], some in the form of series [11, Sect. 3.1.2], but none meets the criteria for a preconditioning BIO stipulated above. 1.3. Prior Work The very same challenge is encountered in two dimensions where the role of Da is played by a straight line segment. In this situation, one of the authors jointly with J.-C. Nédélec managed to find the exact variational inverses of the weakly singular and hypersingular BIOs [12]. Their applicability as explicit Calderón-type preconditioners for open arcs was later shown in [8]. The breakthrough of [12] in 2D provided a boost for the search for preconditioning operators on flat disks. First results were published in [11,13], but 1 The. ±1/2 (Γ) are also denoted as H ±1/2 (Γ) [9]. Sobolev spaces H 00.

(3) IEOT. Inverses of BIOs on Disks. Page 3 of 14. 4. despite some theoretical progress, genuine preconditioning BIOs for V and W remained elusive. 1.4. Main Results We provide the first characterization of the inverses of the hypersingular and the weakly singular BIOs on the unit disk as integral operators on Da with weakly singular/hypersingular kernels in closed form, see Sect. 3.1, (3.1), (3.2) and Sect. 3.2, (3.3), (3.4). We call these new BIOs modified weakly singular and modified hypersingular operators, respectively, because their kernels feature the same structure as those of the standard BIOs, except for an additional smooth cut-off function that depends on the distance to the boundary of the disk ∂Da . However, the modified hypersingular operator, to be read as a finite part integral, fails to meet criterion (i) for a preconditioning operator. We remedy this by providing an equivalent variational form in suitable trace spaces in Sect. 4, Theorem 4.2. It turns out that this variational form is related to the modified weakly singular operator in exactly the same way as the weak form of the standard hypersingular operator can be expressed through that for the weakly singular operator [14, Thm. 6.17]. Instrumental in the derivation of this variational form of the inverse of the weakly singular operator have been recent results by Nédélec [15] also elaborated in the PhD thesis of Ramaciotti [11]. They use so-called projected spherical harmonics in order to state series expansions for the kernels of the boundary integral operators and their inverses on the disk. We make use of such relations and prove them in a simple way in Sect. 4.. 2. Preliminaries 2.1. Notations Let d = 2, 3. For a bounded domain K ⊆ Rd , C m (K), m ∈ N0 , denotes the space of m-times differentiable scalar functions on K, and, similarly, for the space of infinitely differentiable, scalar continuous functions we write C ∞ (K). Let Lp (K) designate the class of p-integrable functions over K. Dual spaces are defined in standard fashion with duality products denoted by angular brackets · , ·K . Let O ∈ Rd be open and s ∈ R. We denote standard Sobolev spaces by s H (O). For positive s and O Lipschitz such that O Õ for Õ ∈ Rd closed, s (O) be the space of functions whose extension by zero to Õ belongs to let H H s (Õ), as in [12]. In particular, the following L2 (O)-duality relations hold −1/2 (O) ≡ H 1/2 (O) 1/2 (O) . H and H −1/2 (O) ≡ H (2.1) 2.2. Variational BIEs on the Disk Da We focus on the circular disk Da with radius a > 0, defined as Da := {x ∈ R3 : x3 = 0 and x < a}. Often, we will omit the third coordinate and use the following short polar coordinate notation: x = (rx cos θx , rx sin θx ) ∈ Da ..

(4) 4. Page 4 of 14. R. Hiptmair et al.. IEOT. 2.2.1. Weakly Singular Integral Equation. As in (1.1), we consider the following singular integral equation: for g ∈ H 1/2 (Da ), we seek a function −1/2 (Da ) such that σ∈H 1 σ(x) (V σ)(y) := dDa (x) = g(y), y ∈ Da . (2.2) 4π Da x − y The measure dDa (x) denotes the surface element in terms of x ∈ Da , equal to rx drx dθx , and the unknown σ is the jump of the Neumann trace of the solution U of the exterior Dirichlet problem for −Δ in R3 \ Da with Dirichlet data g imposed on Da [2, Sect. 3.5.3]. −1/2 (Da ) The symmetric variational formulation for (2.2) is: find σ ∈ H 1/2 such that for g ∈ H (Da ), it holds σ(x)ψ(y) 1 V σ , ψDa = dDa (x)dDa (y) = g , ψDa , (2.3) 4π Da Da x − y −1/2 (Da ) [2, Sect. 3.5.3]. for all ψ ∈ H 2.2.2. Hypersingular Integral Equation. As in (1.1), we consider the following singular integral equation: for μ ∈ H −1/2 (Da ), we seek a function 1/2 (Da ) such that u∈H 1 1 − u(x) (W u)(y) = y ∈ Da , (2.4) 3 dDa (x) = μ(y), 4π Da x − y where the unknown u is the jump of the Dirichlet trace of the solution U of the exterior Neumann problem for −Δ in R3 \Da [2, Sect. 3.5.3] and Neumann data μ. Let v be a continuously differentiable function over Da , and let ṽ be an appropriate smooth extension of v into a three-dimensional neighborhood of Da . Then the vectorial surface curl operator is [14, p.133] curlDa v(x) := n(x) × ∇ṽ(x),. (2.5). with n(x) being a normal at Da in x ∈ Da , and ∇ denoting the standard gradient. This operator is instrumental for stating a variational form of W with a merely weakly singular kernel. curlDa u(y) · curlDa v(x) 1 dDa (x)dDa (y), (2.6) W u , vDa = 4π Da Da x − y 1/2 (Da ). The proof follows the same steps as that of [14, for u, v ∈ H 1/2 (Da ), when integrating by Thm. 6.17] for closed surfaces. Since u, v ∈ H parts, the boundary term vanishes and [14, Lemma 6.16] still holds. Existence and uniqueness of solution of variational problems involving the bilinear forms induced by V and W was proved by Stephan in [4, Thm. 2.7]. Moreover, for a screen Γ, those bilinear forms are continuous and 1/2 (Γ), respectively (cf. [2, Thm. 3.5.9]). One can −1/2 (Γ) and H elliptic in H show that in these cases and for sufficiently smooth screens Γ, when approaching the edges ∂Γ, the solutions decay like the square-root of the distance to ∂Γ [16]..

(5) IEOT. Inverses of BIOs on Disks. Page 5 of 14. 4. 3. New Boundary Integral Operators 3.1. Modified Weakly Singular Integral Operator We define the modified weakly singular operator through the improper integral 2 Sa (x, y) dDa (y), x ∈ Da , (Vυ)(x) := − 2 υ(y) (3.1) π Da x − y with the bounded function on Da × Da ⎛ ⎞ a2 − rx2 a2 − ry2 ⎜ ⎟ Sa (x, y) := tan−1 ⎝ (3.2) ⎠ , x = y. a x − y We remark that the standard weakly singular BIO (as defined in (2.2)) is given by (3.1) without Sa (x, y) and scaled by π2 . Furthermore, since lim Sa (x, y) = π2 when x, y ∈ Da , the kernels of V and V have the same. x→y. weakly singular behavior in (the interior of) Da . Also note that Sa (x, y) = 0 if |x| = a or |y| = a. As a consequence, Sa , though bounded, will be discontinuous on ∂Da × ∂Da . 3.2. Modified Hypersingular Integral Operator We define the modified hypersingular operator W through the finite part integral 2 (3.3) (Wg)(x) := − 2 − g(y)KW (x, y)dDa (y), x ∈ Da , π Da with KW (x, y) :=. a Sa (x, y) √ √ + 3, 2 2 2 2 x − y x − y a − rx a − ry 2. Since lim Sa (x, y) = x→y. π 2. x = y.. (3.4). when x, y ∈ Da , the kernel of the standard. hypersingular operator W (defined as in (2.4)) and the second term in (3.4) −3 have the same O(x − y )-type singularity in the interior of Da . On the other hand, the first term in (3.4) represents a strongly singular kernel in the interior of Da and features a hypersingular behavior when x = y ∈ ∂Da . 3.3. Inverses of V and W The next fundamental result reveals why we are interested in these exotic looking integral operators (cf. [12, Prop. 3.6] or [8, Thm. 2.2]). Theorem 3.1. The integral operators V and W can be extended to continuous operators 1/2 (Da ) and W : H 1/2 (Da ) → H −1/2 (Da ). (3.5) V : H −1/2 (Da ) → H The extensions provide inverses of V and W on trace spaces: W V = IdH −1/2 (Da ) ,. V W = IdH 1/2 (Da ) ,. (3.6). V W = IdH 1/2 (Da ) ,. W V = IdH −1/2 (Da ) .. (3.7).

(6) 4. Page 6 of 14. R. Hiptmair et al.. IEOT. Before we proceed to the proof of Theorem 3.1 we need to show some auxiliary results. We begin by defining the function p(ρ, θ) as p(ρ, θ) :=. ∞ 1 − ρ2 1 1 |n| inθ , ρ e = 2π n=−∞ 2π 1 + ρ2 − 2ρ cos θ. ∀ |ρ| < 1,. (3.8). with θ ∈ [0, 2π] (cf. [17, Chap. 1.1]). Lemma 3.2. When μ is continuosly differentiable, the solution of the hypersingular integral equation (2.4) can be written as u(x) = (Vμ)(x), for all x ∈ Da . Proof. From [18, Thm. 2], we know that if μ is continuously differentiable, then the solution to (2.4) can be written as r r a p xs2 y , θx − θy 4 ds dDa (y). (3.9) u(x) = − − μ(y)− 2 2 1/2 (s2 − r 2 )1/2 π Da max(rx ,ry ) (s − rx ) y For a < ∞, we apply [17, Eq. 1.2.18] and get ⎛ ⎞ 2 − r2 2 − r 2 a s s x y ⎟ −2 μ(y) ⎜ u(x) = 2 − tan−1 ⎝ dDa (y) ⎠ π Da x − y s x − y max(rx ,ry ) ⎛ μ(y) −2 ⎜ = 2− tan−1 ⎝ π Da x − y. a2 − rx2. a2 − ry2. a x − y. ⎞. ⎟ −1 ⎠ − tan (0) dDa (y). −2 Sa (x, y) = 2 − μ(y) dDa (y), π Da x − y with Sa from (3.2).. . Lemma 3.3. Let a > 0 and x, y ∈ Da . If a ≥ s ≥ max(rx , ry ), we find the following primitive r r s2 p xs2 y , θx − θy ds (s2 − rx2 )3/2 (s2 − ry2 )3/2 √ √ 2 2 ⎞ 2 s2 −rx s −ry tan−1 sx−y ⎟ s 1 ⎟. √ √ + =− 3 ⎠ 2π x − y2 s2 − rx2 s2 − ry2 x − y (3.10) Proof. This can be shown by direct calculation using the following change of variable2 : √ √ s4 − rx2 ry2 dη s2 − rx2 s2 − ry2 , = , η := s ds ηs3 2 One. uses the same change of variable to prove [17, Eq. 1.2.18]. To the authors’ knowledge, however, it has never been used for this expression before..

(7) IEOT. Inverses of BIOs on Disks. Page 7 of 14. 4. which leads to . 1 − rx2 ry2 s2 ds (s2 − rx2 )3/2 (s2 − ry2 )3/2 1 + rx2 ry2 − 2rx ry cos(θx − θy ) η −2 = dη, 2 x − y + η 2. where . η −2 2. x − y + η 2. dη = −. 1 2. η x − y. −. tan−1. . η x−y 3. x − y. .. (3.11) . By definition of η the result follows.. Lemma 3.4. The solution of the weakly singular BIE (2.2) can be written as σ(x) = (Wg)(x), for all x ∈ Da , if g is continuously differentiable. Proof. From [18, Thm. 1], we know that if g is continuously differentiable, then the solution to (2.2) can be written as r r a s2 p xs2 y , θx − θy 4 ds dDa (y). (3.12) σ(x) = − g(y)− 2 2 3/2 (s2 − r 2 )3/2 π Da max(rx ,ry ) (s − rx ) y Now, let r̄ := max(rx , ry ). For a < ∞, we use Lemma 3.3 to deal with the inner integral and get. a − r̄. s2 p (s2 −. rx ry. , θ − θy. x s2 rx2 )3/2 (s2 −. . ry2 ). ds = − 3/2. 1 fp 2π tan−1. . x − y2 √. +. 2 s2 −rx. √. s2 −ry2. sx−y. x − y3 . = lim f p →0. . x − y2. s √ s2 − rx2 s2 − ry2 ⎞. √. ⎟a ⎟ ⎟ ⎟ ⎠ r̄. a Sa (x, y) √ + x − y3 a2 − rx2 a2 − ry2. s √ √ 2 x − y s − rx2 s2 − ry2 ⎞ √ 2 2 ⎞ 2 s2 −rx s −ry −1 tan ⎟ ⎟ sx−y ⎟ ⎟ ⎟ ⎟. + 3 ⎟ ⎟ x − y ⎠ s=r̄+ ⎠. −. 2. Here the finite part (fp) of the last expression needs to be considered. This means that we keep the first two terms, drop the third one as it tends to ∞ for → 0. The fourth term vanishes in the limit. Hence, we arrive at a Sa (x, y) 2 √ √ + dDa (y) σ(x) = − 2 − g(y) 3 2 π Da x − y x − y a2 − rx2 a2 − ry2 2 = − 2 − g(y)KW (x, y)dDa (y), π Da.

(8) 4. Page 8 of 14. R. Hiptmair et al.. IEOT. . with KW from (3.4) as stated.. Proof of Theorem 3.1. Both identities in (3.6) follow from Lemma 3.4 combined with the density of C ∞ (Da ) in H −1/2 (Da ). Analogously, the relations in (3.7) are consequences of Proposition (3.2) and the density of C ∞ (Da ) in H 1/2 (Da ). Thus, the mapping properties of W and V are induced by those of V and W. We mention a simple consequence of continuity and ellipticity of the standard BIOs W and V combined with Theorem 3.1. Corollary 3.5. The bilinear forms: (ϑ, μ) → Vϑ , μ D , a (u, g) → Wu , g D , a. ϑ, μ ∈ H −1/2 (Da ), u, g ∈ H. 1/2. (Da ),. (3.13) (3.14). are elliptic and continuous in H −1/2 (Da ) and H 1/2 (Da ), respectively.. 4. Bilinear Form for the Modified Hypersingular Integral Operator Over Da We note that formula (3.4) cannot be used directly when implementing a Galerkin BE discretization. For the hypersingular operator W this flaw was cured by the availability of the variational form (2.6). In this section we establish an analogous relation between W and V. Let us begin by considering the modified weakly singular operator V over the unit disk given by (3.1) and denote ω(x) := 1 − rx2 , x ∈ D1 . Lemma 4.1. The bilinear form associated to the modified hypersingular operator W over D1 satisfies S1 (x, y) 2 curlD1 u(x) · curlD1 v(y)dD1 (x)dD1 (y), Wu , v D1 = 2 π D1 D1 x − y (4.1) 1/2 for all u, v ∈ H∗ (D1 ) := {v ∈ H 1/2 (D1 ) : v , ω −1 D = 0}. 1. This result was first reported by Nedéléc and Ramaciotti in their spectral study of the BIOs over D1 and their variational inverses [11, Def. 2.7.12]. For the sake of completeness, we introduce the key tools they derived and provide a simple proof of this proposition later on. A proof by means of formal integration by parts remains elusive, as it encounters difficulties due to the finite part integrals involved in the definition of W and its kernel introduced in (3.4). It is important to observe that the right-hand side of (4.1) vanishes for constant u or v and, thus, has a non-trivial kernel, if considered in the whole 1/2 H 1/2 (D1 ) space. In other words, the bilinear form in (4.1) is H∗ (D1 )-elliptic 1/2 but does not have this property on H (D1 ). For this reason, its extension to H 1/2 (D1 ) does not actually match the bilinear form of W there, which is H 1/2 (D1 )-elliptic. In order to remedy this situation, we add an appropriate.

(9) IEOT. Inverses of BIOs on Disks. Page 9 of 14. 4. 1/2. regularizing term coming from the definition of H∗ (D1 ) and the known result [10, Eq. (12)] (W1)(y) =. 4 −1 ω (y), π. y ∈ D1 .. From this, we see that for uc constant, (Wuc )(y) evaluates to 2 (Wuc )(y) = 2 uc ω −1 (x)ω −1 (y)dD1 (x), y ∈ D1 , π D1 since 1 , ω −1 D = 2π. Therefore, the following result holds:. (4.2). (4.3). 1. Theorem 4.2. The symmetric bilinear form associated to the modified −1/2 (D1 ) can be written hypersingular operator W : H 1/2 (D1 ) → H as S1 (x, y) 2 curlD1 ,x u(x) · curlD1 ,y v(y)dD1 (x)dD1 (y) Wu , v D = 2 1 π D1 D1 x − y u(x)v(y) 2 dD1 (x)dD1 (y), + 2 (4.4) π D1 D1 ω(x)ω(y) ∀u, v ∈ H 1/2 (D1 ).. Proof. Note that we have added the required regularization to (4.1) such that (4.2) is preserved. This guarantees that our bilinear form defined in (4.4) is by construction equivalent to the bilinear form arising from our modified hypersingular operator W (3.3) on H 1/2 (D1 ). Thus, it is H 1/2 (D1 )-continuous and elliptic. Theorem 4.2 gives us a variational form for W that can be easily implemented. Observe that the chosen regularization to extend the bilinear form 1/2 (4.1) from H∗ (D1 ) to H 1/2 (D1 ) is analogous to the one needed for the modified hypersingular operator on a segment [8, Eq. (2.11)]. In addition, by linearity of the scaling map Ψa : D1 → Da , we get the corresponding relationship on Da : Corollary 4.3. For a > 0, the symmetric bilinear form associated to the mod −1/2 (Da ) can be written as ified hypersingular operator W : H 1/2 (Da ) → H a (x,y) Wu , v D = π22 Da Da Sx−y curlDa ,x u(x) · curlDa ,y v(y)dDa (x)dDa (y) a dDa (x)dDa (y), ∀u, v ∈ H 1/2 (Da ), + aπ2 2 Da Da ωu(x)v(y) a (x)ωa (y) where ωa (x) := a2 − rx2 , x ∈ Da .. (4.5). Our proof of Lemma 4.1 relies on “diagonalization”, the explicit characterization of the eigenfunctions of the BIOs. We begin by introducing key definitions and results on the unit disk D1 . When needed, we provide a short proof for the results reported in [11, Chap. 2]..

(10) 4. Page 10 of 14. R. Hiptmair et al.. IEOT. Definition 4.4. (Projected Spherical Harmonics (PSHs) on D1 [11, Def. 2.7.4]) For l, m ∈ N0 such that |m| ≤ l and x = (rx , θx ), we introduce the functions: (2l + 1) (l − m)! m m imθx m m m . yl (x) := γl e Pl ( 1 − rx2 ), γl := (−1) 4π (l + m)! (4.6) If l1 + m1 and l2 + m2 have the same parity, the following orthogonality holds [11, Prop. 2.7.4] ylm1 1 (y)ylm2 2 (y) 1 m2 dD1 (y) = δll12 δm , (4.7) 1 ω(y) 2 D1 where δll12 is the Kronecker symbol and again ω(y) =. 1 − ry2 .. Lemma 4.5. [19] The PSHs solve a generalized eigenvalue problem for V over D1 in the sense that m 1 y y m (x), l + m even, (4.8) (x) = λm V l ω 4 l l l−m+1 Γ l+m+1 Γ 2 2 m , and Γ being the Gamma function. with λl := l+m+2 l−m+2 Γ Γ 2 2 Lemma 4.6. [20] The PSHs solve a generalized eigenvalue problem for W over D1 in the sense that 1 y m (x) (W ylm )(x) = m l , l + m odd, (4.9) λl ω(x) with λm l as in Proposition 4.5. Remark 4.7. Propositions 4.5 and 4.6 nicely show how, in the case of a disk, the usual V and W have reciprocal symbols but W V = 14 Id due to their mapping properties. This is characterized here by the parity of the PSHs involved. Proposition 4.8. The sets of functions ⎧ 1/2 ⎫ ⎫ ⎧ m (D1 ) ⎪ H {yl : l + m odd} ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1/2 ⎬ ⎬ ⎨ m {yl : l + m even} H (D1 ) . span a dense subspace of m −1 −1/2 : l + m odd} ⎪ {yl ω (D1 ) ⎪ H ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −1/2 ⎭ ⎭ ⎩ m −1 : l + m even} {yl ω H (D1 ) Proof. Since the bilinear form associated to W is symmetric, elliptic and 1/2 (D1 ). Then, the proof continuous, it induces an energy inner product on H 1/2 (D1 ): of the first assertion boils down to showing that for any u ∈ H m (u, ylm )H 1/2 (D1 ) Wu , yl D1 = 0,. ∀ l + m odd. ⇔. Using the symmetry of the bilinear form and (4.9), we derive $ % 1 1 ylm m Wu , yl D1 = m u , = m (u, ylm )1/ω , λl ω D1 λl. u ≡ 0..

(11) IEOT. Inverses of BIOs on Disks. . where (u, v)1/ω :=. D1. Page 11 of 14. 4. u(x)v(x)ω(x)−1 dD1 (x),. is the inner product corresponding to the weighted L2 -space L21/ω (D1 ) [11, Def. 2.7.9]. From [11, Prop. 2.7.9] we know that {ylm : l + m odd} is an orthogonal basis for L21/ω (D1 ). This implies that (u, ylm )1/ω = 0 if and only if u ≡ 0, giving the desired density result. The second assertion follows analogously while the third and fourth can be inferred by the same steps but using the inner product of L2ω (D1 ) [11, Def. 2.7.9] and its corresponding orthogonal bases [11, Prop. 2.7.8]. Remark 4.9. One can prove that odd PSHs can be factored as ylm (x) = eimθx ω(x)Ψ(x),. l + m odd,. where Ψ is a polynomial function-combine [21, Eq. 14.3.21], [22, Eq. 3.2(7)], and [22, Eq. 10.9(22)]. However, this is not true when l + m is even. In that case, the radial part of ylm is already a polynomial (since [22, 10.9(21)] holds instead of [22, 10.9(22)]). This property confirms that the basis functions of our four fractional Sobolev spaces have the correct behaviour. Namely, when 1/2 (D1 ); when l + m is odd, ylm ∼ ω near the boundary and belongs to H −1/2 (D1 ); while the basis l + m is even, ylm ω −1 ∼ ω −1 near ∂D1 and lies in H 1/2 −1/2 functions of H (D1 ) and H (D1 ) have no singular behaviour. We consider the differential operators L+ and L− on D1 [11, Eq. (2.157)] defined as 1 ∂u ∂u +i (4.10) L± u := e±iθ ± , u ∈ C ∞ (D1 ). ∂r r ∂θ Lemma 4.10. ([11, Prop. 2.7.7, Cor. 2.7.2, Lemma 2.7.3]) We have the adjoint relationships L∗+ = L− , and L∗− = L+ . Moreover, for u, v ∈ C ∞ (D1 ) and x, y ∈ D1 holds 1 curlD1 ,x u(x) · curlD1 v(y) = − L+ u(x) L− v(y) + L− u(x) L+ v(y) , 2 (4.11) When applied to PSHs the differential operators L+ and L− yield ym (4.12) L+ ylm−1 = (l − m)(l + m + 1) l , ω m y (4.13) L− ylm+1 = (l + m)(l − m + 1) l . ω Proof of Lemma 4.1. We begin our proof by inferring from W = V−1 , and (4.8) that 4 ym (4.14) Wylm = m l , for l + m even, λl ω Next, we rely on the following recursion formula ' 4 1& (l + m)(l − m + 1)λm−1 , (4.15) = + (l − m)(l + m + 1)λm+1 l l m λl 2.

(12) 4. Page 12 of 14. R. Hiptmair et al.. IEOT. which can be verified by direct computations using the multiplicative property of the Gamma function, i.e., Γ(z + 1) = zΓ(z). Plugging this recursion formula into (4.14) gives 4 ylm λm ω l ' m 1& m+1 yl = (l + m)(l − m + 1)λm−1 . + (l − m)(l + m + 1)λ l l 2 ω (4.16). Wylm =. Then we plug in (4.12) and (4.13). Next, from the fact that W−1 = V, cf. Theorem 3.1, and (4.9), we conclude that ylm±1 = λm±1 ylm±1 , l + m ± 1 odd. (4.17) l ω Then, combining all these ingredients, it is clear that, for (l, m) = (0, 0), l+m even,3 our expression is equivalent to 1 (4.18) L+ V L− ylm + L− V L+ ylm . Wylm = 2 It follows that the associated bilinear form is m1 m2 Wyl1 , yl2 D = 12 L+ V L− ylm1 1 , ylm2 2 D + L− V L+ ylm1 1 , ylm2 2 D 1 1 1 m1 m2 m1 m2 1 = 2 V L− yl1 , L− yl2 D + V L+ yl1 , L+ yl2 D , V. 1. 1. (4.19) for (l1 , m1 ) = (0, 0) and (l2 , m2 ) = (0, 0). Finally, (4.11) and L± = − L∓ imply that (4.19) can be rewritten as the desired formula. It is worth noticing that the condition (l, m) = (0, 0) only excludes the constants represented by y00 . Due to the orthogonality (4.7), this space is defined by 1/2 H∗ (D1 ) = {v ∈ H 1/2 (D1 ) : v , ω −1 D1 = 0}, as introduced in Proposition 4.1. We conclude this proof by extending the obtained formula from {ylm : 1/2 (l, m) = (0, 0), l+m even} to H∗ (D1 ) using linearity and the density results from Lemma 4.8. . 5. Conclusion We have introduced new modified hypersingular and weakly singular integral operators that supply the exact inverses for the standard ones on disks. Moreover, we provide explicit variational forms for these modified BIOs with 3 In. the case (l, m) = (0, 0), (4.15) reads. 4 4 1 1 (0)(1)λ−1 lim Γ(s)s = = 0 + (0)(1)λ0 = 0 λ0 2 π s→0. 4 . In order to go from (4.15) to (4.18), one actually “splits” this limit and therefore breaks π the identity..

(13) IEOT. Inverses of BIOs on Disks. Page 13 of 14. 4. weakly singular kernels that are amenable to standard Galerkin boundary element discretization. The disk Da can serve as a parameter domain for more general screens Γ. Thus the new inverses that we have established on Da are suitable for Calderón preconditioning on three-dimensional parametrized screens, in the same fashion as in 2D [8]. The application of Theorem 3.1 to construct hindependent preconditioners is presented in [23,24] focusing on the numerical use of the new modified BIOs. Though we have developed our analysis only for the Laplace equation over disks with Dirichlet and Neumann boundary conditions, extensions to other elliptic equations, like the scalar Helmholtz equation, can be immediately achieved via perturbation arguments.. References [1] McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000) [2] Sauter, S., Schwab, C.: Boundary Element Methods, Springer Series in Computational Mathematics, vol. 39. Springer, Heidelberg (2010) [3] Stephan, E.P.: A boundary integral equation method for three-dimensional crack problems in elasticity. Math. Methods Appl. Sci. 8, 609–623 (1986) [4] Stephan, E.: Boundary integral equations for screen problems in R3 . Integral Equ. Oper. Theory 10, 236–257 (1987) [5] Buffa, A., Christiansen, S.H.: The electric field integral equation on Lipschitz screens: definitions and numerical approximation. Numer. Math. 94, 229–267 (2003) [6] Steinbach, O., Wendland, W.: The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math. 9, 191–216 (1998) [7] Hiptmair, R.: Operator preconditioning. Comput. Math. Appl. 52, 699–706 (2006) [8] Hiptmair, R., Jerez-Hanckes, C., Urzúa-Torres, C.: Mesh-independent operator preconditioning for boundary elements on open curves. SIAM J. Numer. Anal. 52, 2295–2314 (2014) [9] Lions, J., Magenes, F.: Nonhomogeneous Boundary Value Problems and Applications. Springer, Berlin (1972) [10] Martin, P.A.: Exact solution of some integral equations over a circular disc. J. Integral Equ. Appl. 18, 39–58 (2006) [11] Ramaciotti Morales, P.: Theoretical and numerical aspects of wave propagation phenomena in complex domains and applications to remote sensing. PhD Thesis, Université Paris-Saclay (2016) [12] Jerez-Hanckes, C., Nédélec, J.: Explicit variational forms for the inverses of integral logarithmic operators over an interval. SIAM J. Math. Anal. 44, 2666– 2694 (2012) [13] Ramaciotti, P., Nédélec, J.-C.: About some boundary integral operators on the unit disk related to the Laplace equation. SIAM J. Numer. Anal. 55, 1892–1914 (2017) [14] Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements. Springer, New York (2008).

(14) 4. Page 14 of 14. R. Hiptmair et al.. IEOT. [15] Nédélec, J.-C.: About some operators on the unit disc related to the Laplacian equation. In: Journées Singulières Augmentées (2013). https://jsa2013. sciencesconf.org/conference/jsa2013/T52.pdf [16] Costabel, M., Dauge, M., Duduchava, R.: Asymptotics without logarithmic terms for crack problems. Commun. Partial Differ. Equ. 28, 869–926 (2003) [17] Fabrikant, V.I.: Mixed Boundary Value Problems of Potential Theory and Their Applications in Engineering, Mathematics and its Applications, vol. 68. Kluwer, Dordrecht (1991) [18] Li, X.-F., Rong, E.-Q.: Solution of a class of two-dimensional integral equations. J. Comput. Appl. Math. 145, 335–343 (2002) [19] Wolfe, P.: Eigenfunctions of the integral equation for the potential of the charged disk. J. Math. Phys. 12, 1215–1218 (1971) [20] Krenk, S.: A circular crack under asymmetric loads and some related integral equations. J. Appl. Mech. 46, 821–826 (1979) [21] NIST Digital library of mathematical functions. http://dlmf.nist.gov/, Release 1.0.10 of 2015-08-07 [22] Bateman, H., Erdélyi, A.: Higher Transcendental Functions. McGraw-Hill, New York (1952) [23] Hiptmair, R., Jerez-Hanckes, C., Urzúa-Torres, C.: Optimal operator preconditioning for hypersingular operator over 3D screens. Tech. Rep. 2016-09, Seminar for Applied Mathematics, ETH Zürich, Switzerland (2016). https://www. sam.math.ethz.ch/sam reports/reports final/reports2016/2016-09 rev1.pdf [24] Hiptmair, R., Jerez-Hanckes, C., Urzúa-Torres, C.: Optimal Operator Preconditioning for weakly singular Operator over 3D screens Tech. Rep. 2017-13, Seminar for Applied Mathematics, ETH Zürich, Switzerland (2017). https:// www.sam.math.ethz.ch/sam reports/reports final/reports2017/2017-13 fp.pdf R. Hiptmair (B) and C. Urzúa-Torres Seminar for Applied Mathematics ETH Zurich Raemistrasse 101 8092 Zurich Switzerland e-mail: ralf.hiptmair@sam.math.ethz.ch C. Urzúa-Torres e-mail: carolina.urzua@sam.math.ethz.ch C. Jerez-Hanckes School of Engineering Pontificia Universidad Católica de Chile Av. Vicuña Mackenna 4860 Macul Santiago Chile e-mail: cjerez@ing.puc.cl Received: March 24, 2017. Revised: October 29, 2017..

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