Double Layer Potentials on Polygons and Pseudodifferential Operators on Lie Groupoids

Transcription

1 Integr.Equ.Oper.Theory (18)9:14 c Springer International Publishing AG, part of Springer Nature 18 Integral Equations and Operator Theory Double Layer Potentials on Polygons and Pseudodifferential Operators on Lie Groupoids Yu Qiao and Hengguang Li Abstract. We use an approach based on pseudodifferential operators on Lie groupoids to study the double layer potentials on plane polygons. Let be a simply connected polygon in R. Denote by K the double layer potential operator on associated with the Laplace operator Δ. We show that the operator K belongs to the groupoid C -algebra that the first named author has constructed in an earlier paper (Carvalho and Qiao in Cent Eur J Math 11(1):7 54, 13). By combining this result with general results in groupoid C -algebras, we prove that the operators ±I + K are Fredholm between appropriate weighted Sobolev spaces, where I is the identity operator. Furthermore, we establish that the operators ±I + K are invertible between suitable weighted Sobolev spaces through techniques from Mellin transform. The invertibility of these operators implies a solvability result in weighted Sobolev spaces for the interior and exterior Dirichlet problems on. Mathematics Subject Classification. Primary 45E1, 58H5; Secondary 47G4, 47L8, 47C15, 45P5. Keywords. Double layer potential operators, Pseudodifferential operators on Lie groupoids, Groupoid C -algebras, Weighted Sobolev spaces, Mellin transform. 1. Introduction Potential theory can be traced back to the works of Lagrange, Laplace, Poisson, Gauss, and others [4], and plays a fundamental role in many real-world problems, especially in physics. Many works are devoted to the method of layer potentials. We mention here a few monographs, beginning with the books by Courant and Hilbert [14], Folland [1], Hsiao and Wendland [3], Qiao was partially supported by the NSFC Grant and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. Li was partially supported by the NSF Grant DMS , by the Natural Science Foundation of China (NSFC) Grant , and by the AMS Fan China Exchange Program.

2 14 Page of 6 Y. Qiao, H. Li IEOT Kress [3], Mclean [4], and Taylor [69]. These monographs give a rather complete account of the theory of layer potential operators on smooth domains. Let us also mention the paper [], which includes some results on C 1 -domains. There are also many papers devoted to the method of layer potentials on non-smooth domains, which can be roughly divided into two categories: one devoted to Lipschitz domains and the other to polyhedral domains. The case of Lipschitz domains, by far the most studied among the class of non-smooth domains, is also fairly well understood. We mention the papers of Jerrison and Kenig [4, 5], Kenig [8], and Verchota [7] for relevant results on domains in the Euclidean space. In the works of Mitrea and Mitrea [47], Mitrea and Mitrea [49], Mitrea and Taylor [51,5], and Kohr et al. [9], the method of layer potentials is applied to Lipschitz domains on manifolds. See also Costabel s paper [1] for an introduction to the method of layer potentials, in which more elementary methods are applied. We are interested in nonsmooth domains, especially in polyhedral domains. By comparison, much fewer works were dedicated to this case. We mention however the papers of Ammann et al. [], Lewis and Parenti [35], and Mitrea [48] for results on polygonal domains. The works of Elschner [18], Fabes et al. [19], Angell et al. [6], Medkova [43], and Verchota and Vogel [71] deal with the case of polyhedral domains in three and four dimensions. The paper [41] concentrates on polyhedral domains and domains with cracks. See [7] for the related case of interface problems. In addition, boundary value problems on domains with conical points were studied by many authors. We mention in this regard the work of Kondratiev [3], the papers of Kapanadze and Schulze [6], Lewis and Parenti [35], Li et al. [36], Mazzeo and Melrose [4] and Melrose [44], and Schröhe and Schulze [64,65]. See also the books of Egorov and Schulze [17], Kozlov et al. [31], Mazya and Rossmann [39], Melrose [45, 46], Schulze [66], Schulze et al. [67], and Sauter and Schwab [63]. Many of these works are devoted to constructing suitable algebras of pseudodifferential operators on conical manifolds. See also the paper [1,4,5,15,16] using groupoids to construct algebras of pseudodifferential operators on singular spaces, and [58,68] for some related constructions. In this paper, we study the double layer potential operator K associated with the Laplace operator on a plane polygon. Let R be a (regular) open bounded domain. Consider the interior Dirichlet problem { Δu = in u = φ on, (1) and the exterior Dirichlet problem { Δu = in c u = φ on, () where c denotes the complement of, i.e., c = R \.

3 IEOT Double Layer Potentials on Polygons Page 3 of 6 41 For ψ Cc ( ), define the double layer potential (x y) ν(y) u(x) = c x y ψ(y)dσ(y), (x R \ ), where ν(y) is the exterior unit normal to a point y andc is a constant. Conventionally c is taken to be 1 π in this paper. let u (x) andu + (x) denote the limits of u(z) asz x nontangentially from z andz R \, respectively. The classical results [13,1,69] on double layer potentials give that for (a.e.) x, we have 1. u (x) =ψ(x)+kψ(x), i.e., u =(I + K)ψ;. u + (x) = ψ(x)+kψ(x), i.e., u + =( I + K)ψ, where Kψ(x) = k(x, y)ψ(y)dσ(y), (x y) ν(y) with k(x, y) = c x y. Hence, the interior and exterior Dirichlet problems are reduced to solving boundary integral equations (I + K)ψ = φ and ( I + K)ψ = φ, respectively, where φ is the given function on the boundary andψ is the unknown function on. In general, the double layer potential method works for (regular) domains in R n, n. For instance, in [1,69], it is shown that if the domain R n has C boundary, then the double layer potential operator K is compact on L ( ) (and H m ( )). Hence operators ±I + K are Fredholm of index zero. Therefore, the solvability of the interior and exterior Dirichlet problems is equivalent to injectivity or surjectivity of ±I + K. If the boundary is not C, the operator K is no longer compact (see [18,19,1,,3,3,35,47,48,5,7]). However, we can still hope that ±I +K are Fredholm operators on appropriate function spaces on the boundary. Recently, Perfekt and Putinar have studied the essential spectrum of the double layer potential operator K on a planar domain with corners and give a complete result of the essential spectrum of K on the Sobolev space of order 1 along the boundary [59,6]. From the pseudodifferential operator point of view, if the boundary is smooth, the double layer potential operator K is a pseudodifferential operator of order 1 on the boundary [69]. The survey [38] emphasizes the importance of understanding the algebra of pseudodifferential operators on singular spaces. In this paper, we use a groupoid approach to construct algebras of pseudodifferential operators (and C -algebras) on polygons in the spirit of [3, 56], specifically in the framework of Fredholm groupoids [9, 1]. Then we show that the double layer potential operator K lies in this groupoid C -algebra. From this result, we demonstrate that the operators ±I + K are Fredholm between appropriate weighted Sobolev spaces on the boundary of the domain. Consequently, we use techniques from Mellin transform to prove that the operators ±I + K are isomorphic between suitable weighted Sobolev spaces. This implies a solvability result in weighted Sobolev spaces for the

4 14 Page 4 of 6 Y. Qiao, H. Li IEOT interior and exterior Dirichlet problems on. It is also possible to extend our method to solve interior and exterior Neumann problems. Our main results are as follows. Let be a simply connected polygon in R with vertices P 1,P,...,P n. Denote by θ i the interior angle at vertex P i. Throughout the paper, we always assume that be a simply connected polygon in R. Let K m 1 +a( ) be the Sobolev space on with weight r and index a (see Sect. ). Define θ := min { π θ 1, π π θ 1, π θ, Clearly, 1 <θ < 1. Then we have π π θ,..., π θ n, π π θ n Theorem 1.1. For a ( θ, 1/) and m, the operators ±I + K : K m 1 +a( ) Km 1 +a( ) are isomorphisms. The paper is organized as follows. In Sect., we review weighted Sobolev spaces on plane polygons and briefly recall desingularization of polygons. In Sect. 3, we collect basic concepts of pseudodifferential operators on Lie groupoids. Then, we give an explicit analysis on the double layer potential operator K associated to a plane sector and discuss its connection to a (smooth invariant) family of operators on certain Lie groupoid in Sect. 4. Section 5 contains the proofs our main result. Namely, the operators ±I + K are isomorphisms on weighted Sobolev spaces with suitable weights. We end with concluding remarks in Sect. 6. }.. Weighted Sobolev Spaces on Polygons and Desingularization Let be a plane polygon and m Z.Letα be a multi-index, and r be the weight function which is equivalent to the distance function to the vertices of (see [7] for details). We define the mth Sobolev space on with weight r and index a by Ka m () = {u L loc() r α a α u L (), for all α m}. The norm on Ka m () is u Ka m () := α m r α a α u L (,dx). By Theorem 5.6 in [7], this norm is equivalent to u m,a := r a (r ) α u L (,dx), α m where (r ) α =(r 1 ) α1 (r ) α (r n ) αn. Clearly, we have that rk t a m () = Ka+t(). m

5 IEOT Double Layer Potentials on Polygons Page 5 of 6 41 In general, this isomorphism may not be an isometry. In [7], there is a standard procedure to desingularize. Denote by Σ() the desingularization of, which is a Lie manifold with boundary. The space L (Σ()) is defined by using the volume element of a compatible metric with the Lie structure at infinity on Σ(). A compatible metric is r g e, where g e is the Euclidean metric. Then Sobolev spaces H m (Σ()) are defined by using L (Σ()). Proposition.1. We have, for all m Z, K1 m () = H m (Σ(),g), where the metric g = r g e. Proof. The result follows from Proposition 5.7 in [7]. The identification given above allows us to define weighted Sobolev spaces on the boundary K m a ( ). For more details, see [7]. Proposition.. For m Z, we have the following identification: K m 1 ( ) = H m ( Σ()), where Σ() is the desingularization of and Σ() is the union of hyperfaces which are not at infinity. Proof. The result follows from Definition 5.8 in [7]. Therefore, we have the following identifications for the weighted Sobolev spaces both on and on the boundary. Proposition.3. We have, for all m Z, K1 m () = H m (,g), and K m 1 ( ) = H m (,g), where the metric g = r g e. 3. Pseudodifferential Operators on Lie Groupoids 3.1. Lie Groupoids In this subsection, we review some basic facts on Lie groupoids. We begin with the definition of groupoids. Definition 3.1. A groupoid is a small category G in which each arrow is invertible. Let us make this definition more precise [8,34,37,53,6]. A groupoid G consists of two sets: one of objects (or units) G and the other of arrows G 1. Usually we shall identify G = G 1, denote M := G, and use the notation G M. First of all, to each arrow g Gwe associate two units: its domain d(g) and its range r(g), i.e., d, r : G M. Then we define the set of composable pairs G () := {(g, h) G G d(g) =r(h)}.

6 14 Page 6 of 6 Y. Qiao, H. Li IEOT The multiplication μ : G () G () is given by μ(g, h) =gh, and it is associative. Moreover, we have an injective map u : M G, where u(x) isthe identity arrow of an object x M. The inverse of an arrow is denoted by g 1 = ι(g). We can write (in [53]) G () μ ι d G G M u G. r A groupoid G is therefore completely determined by the sets M, G and the structural maps d, r, μ, u, ι. The structural maps satisfy the following properties: 1. d(hg) =d(g), r(hg) =r(h),. k(hg) =(kh)g 3. u(r(g))g = g = gu(d(g)), and 4. d(g 1 )=r(g), r(g 1 )=d(g), g 1 g = u(d(g)), and gg 1 = u(r(g)) for any k, h, g G 1 with d(k) =r(h)andd(h) =r(g). The following definition is taken from [34]. Definition 3.. A Lie groupoid is a groupoid G =(G, G 1,d,r,μ,u,ι) such that M := G and G 1 are smooth manifolds, possibly with corners, with M Hausdorff, the structural maps d, r, μ, u, andι are smooth and the domain map d is a submersion (of manifolds with corners). Remark 3.3. In general, the space G 1 may not be Hausdorff. However, since d is a submersion, it follows that each fiber G x := d 1 (x) (respectively G x := r 1 (x)) is a smooth manifold without corners, see [34,57], hence it is Hausdorff. Note that the groupoids in this paper will be Hausdorff. 3.. Pseudodifferential Operators and Groupoid C -Algebras We recall briefly the construction of the space of pseudodifferential operators associated to a Lie groupoid G with units M [33,34,54,55,58]. The dimension of M is n 1. Let P =(P x ), x M be a smooth family of pseudodifferential operators acting on G x. We say that P is right invariant if P r(g) U g = U g P d(g), for all g G, where U g : C (G d(g) ) C (G r(g) ), (U g f)(g )=f(g g). Let k x be the distributional kernel of P x, x M. Note that the support of P supp(p ):= supp(k x ) {(g, g ),d(g) =d(g )} G G x M since supp(k x ) is contained in G x G x.letμ 1 (g,g):=g g 1. The family P =(P x ) is called uniformly supported if its reduced support supp μ (P ):= μ 1 (supp(p )) is a compact subset of G. Definition 3.4. The space Ψ m (G) ofpseudodifferential operators of order m on a Lie groupoid G with units M consists of smooth families of pseudodifferential operators P =(P x ), x M, with P x Ψ m (G x ), which are uniformly supported and right invariant.

7 IEOT Double Layer Potentials on Polygons Page 7 of 6 41 We also denote Ψ (G) := m R Ψm (G) and Ψ (G) := m R Ψm (G). We then have a representation π of Ψ (G) oncc (M) (or on C (M), on L (M), or on Sobolev spaces), called the vector representation uniquely determined by the equation (π(p )f) r := P (f r), where f Cc (M) andp =(P x ) Ψ m (G). Remark 3.5. If P Ψ (G), then P identifies with the convolution with a smooth, compactly supported function, hence Ψ (G) identifies with the convolution algebra Cc (G). In particular, we can define k P I,d := sup k P (g 1 ) dμ x (g), k P I,r := sup k P (g 1 ) dμ x (g), x M G x x M G x and P L1 (G) := max{ k P I,d, k P I,r }. The space L 1 (G) is defined to be the completion of Ψ (G) Cc (G) inthe norm L 1 G. For each x M, there is an interesting family of representation of Ψ (G), the regular representations π x on Cc (G x ), defined by π x (P )=P x. It is clear that if P Ψ n 1 (G) π x (P ) L (G x) P L 1. The reduced C norm of P is defined by P r =sup π x (P ) =sup P x, x M x M and the full norm of P is defined by P =sup ρ(p ), ρ where ρ varies over all bounded representations of Ψ (G) satisfying ρ(p ) P L 1 (G) for all P Ψ (G). Definition 3.6. Let G be a Lie groupoid and Ψ (G) be as above. We define C (G) (respectively, C r (G)) to be the completion of Ψ (G) in the norm (respectively, r ). If r =,thatis,ifc (G) = C r (G), we call G metrically amenable. We give some examples of Lie groupoids. Example 3.7 (Manifolds with corners). A manifold (with corners) M may be viewed as a Lie groupoid, by taking both the object and morphism sets to be M, and the domain and range maps to be the identity map M M, and Ψ (M) =Cc (M). Example 3.8 (Lie groups). Every Lie group G can be regarded as a Lie groupoid G = G with space of units M = {e}, the unit of G. And Ψ m (G) is the algebra of properly supported and invariant pseudodifferential operators on G.

8 14 Page 8 of 6 Y. Qiao, H. Li IEOT Example 3.9 (Pair groupoid). Let M be a smooth manifold. Let G = M M G = M, with structure maps d(m 1,m )=m, r(m 1,m )=m 1,(m 1,m )(m,m 3 )= (m 1,m 3 ), u(m) = (m, m), and ι(m 1,m ) = (m,m 1 ). Then G is a Lie groupoid, called the pair groupoid of M. According to the definition, a pseudodifferential operator P belongs to Ψ m (G) if and only if the family P = (P x ) x M is constant. Hence we obtain Ψ m (G) =Ψ m comp(m). Also, an important result is that C (G) = K, the ideal of compact operators, the isomorphism given by the vector representation or by any of the regular representations (together with G x = M). If M is a discrete set with k elements, then C (G) = M k (C) and the convolution product is given by matrix multiplication. Example 3.1 (Transformation (or Action) groupoid). Suppose that a Lie group G acts on the smooth manifold M from the right. The transformation groupoid over M {e} = M, denoted by M G, is the set M G with structure maps d(m, g) = (m g, e), r(m, g) = (m, e), (m, g)(m g, h) = (m, gh), u(m, e) =(m, e), and ι(m, g) =(m g, g 1 ). For more on the action groupoid, one may see [37,53,6]. Example 3.11 (Bundle of Lie groups). If G M is a bundle of Lie groups, i.e, d = r (hence each fiber is a Lie group), then Ψ m (G) consists of smooth families of invariant and properly supported pseudodifferential operators on the fibers of G M. Clearly, vector bundles are a special case of bundle of Lie groups. 4. Double Layer Potentials on Plane Sectors We consider a plane sector θ := {rα : r (, ),α (,θ)} with angle θ. Thus, the boundary θ consists of two rays, which we label as L 1 and L, respectively The Double Layer Potential Operator Associated with a Plane Sector Before we calculate the explicit form of the double layer potential operator associated to the Laplace operator and θ, we recall the definition of the Mellin convolution operator and the definition of the Mellin transform [61]. Definition 4.1. Let p = p(r) Cc (R + )andu Cc (R + ). Define the function Pu on R + by Pu(r) =p u(r) = p(r/s) u(s) ds s. The operator P will be called the smoothing Mellin convolution operator on R + with convolution kernel p. Definition 4.. Let p be the convolution kernel of a smoothing Mellin convolution operator P on R +. The Mellin transform Mp of p is defined by Mp(t) =q(t) = s it p(s) ds s.

9 IEOT Double Layer Potentials on Polygons Page 9 of 6 41 We recall the following standard properties of the Mellin transform [61]. Proposition 4.3. Suppose P is a smoothing Mellin convolution operator with convolution kernel p. Then for any u C c (R + ), we have M(p u)(t) =M(Pu)(t) =Mp(t)Mu(t). Remark 4.4. Some general references on Mellin convolution operators and the Mellin transform in solving boundary value problems, include Kapanadze and Schulze [6], Egorov and Schulze [17], Lewis and Parenti [35], Melrose [45,46], Schrohe and Schulze [64, 65], Schulze [66]. The double layer potential with a function φ on θ is defined by (K θ φ)(x) := 1 (x y) ν(y) π θ x y φ(y)dσ(y), (3) where x, y, ν(y) is the exterior unit normal to a point y θ.sok θ depends on the locations of x and y on the boundary. We further define for i =1,, φ i (x) :=φ(x) for x L i, φ i (x) :=, otherwise. Denote by (K ij φ i )(x) := 1 (x y) ν(y) π L j x y φ i (y)dσ(y), j =1,. Note that if x, y belong to the same ray, then x y is perpendicular to ν(y), i.e., (x y) μ(y) =. Then, it is clear that the operator K θ can be represented as a matrix, i.e., ( )( ) K1 φ1 K θ φ =. K 1 φ Let w and z be the points on L 1 and L, respectively. A direct calculation leads to (K 1 φ )(w) = k θ (w/z)φ (z) dz z, (K 1 φ 1 )(z) = k θ (z/w)φ 1 (w) dw w, where k θ (r) = 1 r sin θ π r +1 rcos θ. (4) It is clear that K 1 and K 1 are both Mellin convolution operators with the same kernel k θ. For simplicity, in the text below, we let k = k θ. Then, for any ϕ Cc (R + )andr R +, we define ( Kϕ)(r) = k(r/s)ϕ(s) ds s. (5) Therefore, the operator K is a convolution operator, and K 1 = K 1 = K on C c (R + ). We need the following lemma.

10 14 Page 1 of 6 Y. Qiao, H. Li IEOT Lemma 4.5. For each ξ R, define cos(xξ) f(ξ) = e x dx. cosθ + e x Then we have f(ξ) = π ( e (π θ)ξ e θξ ) sin θ e πξ. (6) 1 Moreover, for all ξ R, we have <f(ξ) π sin θ. Proof. Since f(ξ) is an even function, we can suppose that ξ is positive. It is easy to see that e izξ f(ξ) = e z + e z cosθ dz, We choose the contour Γ = Γ 1 Γ Γ3 Γ4, where Γ 1 = {(x, ) M x M}, Γ = {(M,iy) y M}, Γ 3 = {(x, Mi) M x M}, and Γ 4 = {( M,iy) y M}, form large enough. On Γ,wehave e izξ Γ e z + e z cosθ dz M = e yξ e M+iy + e M iy cosθ dy M 1 e M e M dy Hence we see that as M, Γ e izξ M e M e M. e z + e z dz. cosθ For the same reason, as M,wehave Γ4 e izξ e z + e z dz. cosθ On Γ 3, if we take M k =kπ, then we have Γ 3 e izξ e z + e z cosθ dz Mk = M k Mk M k e Mkξ e x+im k + e x im k cosθ dx e Mkξ e x + e x cosθ dx e M kξ e x + e x cosθ dx Ce Mkξ, where C is a constant. Thus as M k,wehave Γ3 e izξ e z + e z dz. cosθ

11 IEOT Double Layer Potentials on Polygons Page 11 of 6 41 Now let us find the singularities of the integrand, i.e., the roots of the equation e z + e z cosθ =. So we get z =(kπ ± θ)i, where k =, ±1, ±,... In the interior of Γ, we see that z = θi, (π ± θ)i, (4π ± θ)i,... Let e izξ g(z) = e z + e z cosθ. Next let us compute the residue of g(z) at each pole. It is clear that each singularity is simple. Therefore, we calculate Res(g, (kπ ± θ)i) = e izξ (z (kπ ± θ)i) lim z (kπ±θ)i e z + e z cosθ = e (kπ±θ)ξ i sin(kπ ± θ) = e (kπ±θ)ξ i sin(±θ) Therefore, the Residue Theorem allows us to compute f(ξ) = π ( e θξ + (e (kπ+θ)ξ e (kπ θ)ξ)) sin θ k=1 = π ) (e θξ + e (π+θ)ξ e (π θ)ξ sin θ 1 e πξ 1 e πξ (e θξ eθξ e θξ ) = π sin θ = π sin θ e πξ 1 ( e (π θ)ξ e θξ ), e πξ 1 where we use the fact that the series is absolutely convergent. This proves (6). Define ϕ(θ) =e θξ eθξ e θξ e πξ 1. Then we have ϕ() = 1, ϕ(π) =, and ϕ(π) = 1. Moreover, we compute ϕ (θ) = ξe θξ ξeθξ + ξe θξ e πξ < for any positive ξ>. 1 This implies that ϕ(θ) > for all θ (,π)andϕ(θ) < for all θ (π, π). As a consequence, we have <f(ξ) π for any positive ξ. sin θ Moreover, we have f() = lim f(ξ) = π ξ sin θ π θ = π θ π sin θ >.

12 14 Page 1 of 6 Y. Qiao, H. Li IEOT Since f is even, we see that <f(ξ) π for all ξ R. sin θ Remark 4.6. By the above lemma, the Mellin transform of k(r) in(4) canbe computed as follows Mk(t) = 1 π = 1 π = 1 π = sin θ π s it e itx s sin θ ds s +1 scos θ s e x sin θ e x +1 e x cos θ dx cos(xt)sinθ e x + e x cosθ dx (e θt eθt e θt ) π sin θ e πt 1 = e(π θ)t e θt e πt. 1 Note that the function Mk(z) = e(π θ)z e θz e πz 1 is holomorphic in the strip {z C : 1 < I(z) < 1}, where I(z) isthe imaginary part of z. Let M f denote the multiplication operator by f. Recall the operator K from Equation (5). Then, M r a KMr a has Mellin convolution kernel k a (r) = 1 π r a+1 sin θ r r cos θ +1. Notice that k a (r) is a smooth function on r> (provided that a> 1). The Mellin transform of k a is calculated as follows Mk a (t) = e(π θ)(t ai) e θ(t ai) (7) e π(t ai) 1 = Mk(t ai). The double layer potential operator associated to θ and the Laplace ( ) ( K1 operator takes the form K θ = = K ).Thus,wehave K 1 K ( ) ( ) ( ) Mr a Mr a M K θ,a := K M θ = r a KMr a. r a M r a M r a KMr a Let K a := M r a KMr a. The following theorem is well-known. We give a short proof here. Theorem 4.7. The operator ±I + K θ,a := ±I + M r ak θ M r a is invertible if and only if 1 (M K a (t)) for all t R, wherem K a is the Mellin Transform of K a.

13 IEOT Double Layer Potentials on Polygons Page 13 of 6 41 Proof. The identity operator I has convolution kernel δ(x/y) and the Mellin transform of the δ-function is the function of constant 1. By Proposition 4.3, we know that the Mellin transform of a Mellin convolution operator is the multiplication operator by the Mellin transform of the kernel function. Clearly, K a is a Mellin convolution operator. Then the theorem follows. Recall Proposition.. We have, for all m Z, K m 1 ( θ ) = H m ( θ,g), where the metric g = r g e. We define a θ by a θ = min{ π { θ, π π/(π θ), <θ<π, π θ } = π/θ, π < θ < π. Theorem 4.8. If a ( a θ,a θ ), then ±I + M r ak θ M r a is invertible on ( θ ), that is, K m 1 are invertible. ±I + K θ : K m 1 +a( θ) K m 1 +a( θ) Proof. Recall the Mellin transform of k a in Eq. (7). By Theorem 4.7, we need to find the (positive) smallest a> such that Mk a (t) =±1 for some t R. Therefore, we compute e (π θ)(t ai) e θ(t ai) = ±(e π(t ai) 1) e (π θ)(t ai) ± 1=e θ(t ai) ± e π(t ai) e (π θ)(t ai) ± 1=e θ(t ai) (1 ± e (π θ)(t ai) ) For case +, we have e θ(t ai) =1ore (π θ)(t ai) = 1. For case, we have e θ(t ai) = 1 ore (π θ)(t ai) =1. Hence we obtain θt aθi = kπi or (π θ)t (π θ)ai = kπi, k = ±1, ±,... t = and a = kπ or a = kπ,k= ±1, ±,. θ π θ Hence the (positive) smallest a would be a θ = min{ π θ, π π θ }. Remark 4.9. The interior and exterior Dirichlet problems correspond to the operators I +K θ and I +K θ, respectively. The above calculation shows that the interior and exterior Dirichlet problems are indistinguishable when we use the double layer potentials. So we should consider the operators ±I + K θ at the same time. The following proposition is needed, which gives the explicit description of the function in the kernel of ±I + K θ. Proposition 4.1. Suppose that a function u satisfies that (±I + K θ )u =, then u is of the form

14 14 Page 14 of 6 Y. Qiao, H. Li IEOT u(r) = i c i r ai where c i s are constants, and a i { kπ θ, kπ π θ, k = ±1, ±,...} Proof. Since u satisfies (I + K θ )u =, multiplying both sides by r a from the left gives r a u + r a K θ r a (r a u)= Hence we have (I + r a K θ r a )(r a u)=. By Theorem 4.7 and the above calculation in Theorem 4.8, we know that Mk a (t) =±1 t = and a = kπ θ, kπ,k= ±1, ±,... π θ Therefore, the Mellin transform of the function r a u has support only at t =, so u is a linear combination of r ai s. The case for I + K θ is the same. 4.. Relations to Lie Groupoids We are in position to identify the double layer potential operator K with a smooth invariant family of operators on some Lie groupoid. Let H =[, ] R +, where R + =(, ) is regarded as a commutative group. So H is an action groupoid. It is easy to see that C ( H) =C([, ]) R +. Notice that C[, ] is a unital commutative C -algebra and C ( H) isnot unital. Moreover, we have C (R + )=C (R + ) C ( H). Next we would like to define an (order ) invariant family P on the groupoid H =[, ] (, ), such that π(p )= K, where K is defined by Eq. (5) andπ is the vector representation of Ψ ( H) oncc (, ) uniquely determined by (π(p )f) r = P (f r). We notice that [, ] is the space of units and (, ) is an invariant open dense subset of the compact space [, ]. Then π(ψ ( H)) maps C c (, ) to itself. We define a map φ x : R + H x by φ x (x) =(x x, x 1 ). It is easy to see that the map φ x is a diffeomorphism for all x [, ]. So we can use this map φ x to identify R + and H x. For any f(x) Cc (, ), we define F (x, y) =f(y 1 ), (x, y) H. Clearly the function F (x, y) is smooth on H. Furthermore, if we restrict the function F to H x,wegetf Hx = F (x x, x 1 ). Hence we obtain in this way

15 IEOT Double Layer Potentials on Polygons Page 15 of 6 41 a smooth function on Cc ( H x ). On the other hand, any smooth function on H x can be written (by using φ x )intheformg(x x, x 1 ). Finally, if x [, ], we have a one-to-one correspondence between Cc (, ) and Cc ( H x ) in the following way: f Cc (, ) F Hx = F (x x, x 1 ) Cc ( H x ). Suppose that p(x, y) is a smooth function on (, ) (, ). We can define an integral operator on Cc (, ) by ( Pf)(x) = p(x, y)f(y) dy y, f C c (, ). Then we define p : H x H x R by x (, ) p Hx H x : p((x x, x 1 ), (x y, y 1 )) = p(x x, x y), where we use the map φ x to identify (, ) and H x. We define a family of integral operators P =(P x ), where P x : Cc ( H x ) C ( H x ), x (, ), given by (P x F )(x x, x 1 )=(P x f)(x) = p(x x, x y)f(y) dy y, where f(x) =(F φ x )(x). Lemma The family of integral operators P =(P x ), x (, ), is invariant. Proof. For fixed x 1,x (, ), there exists a unique element g =(x,x 1 x 1) H such that d(g) =d(x,x 1 x 1)=x 1 and r(g) =r(x,x 1 x 1)=x. Suppose we have F Cc ( H x1 ). Then F can be written as F (x 1 x, x 1 )=f(x), where f Cc (, ). So we have (U g F )(x x, x 1 )=F ( (x x, x 1 )(x,x 1 x 1) ) = F (x x, x 1 x 1x 1 ), therefore, (P x U g F )(x x, x 1 )= p(x x, x y)f(x 1 1 x y) dy y. On the other hand, we obtain (P x1 F )(x 1 x, x 1 )= p(x 1 x, x 1 y)f(y) dy y. Let h(x 1 x, x 1 )=(P x1 F )(x 1 x, x 1 ). Thus (U g h)(x x, x 1 )=h(x x, x 1 x 1x 1 ) = p(x x, x 1 y)f(y) dy y = p(x x, x z)f(x 1 1 x z) dz z,

16 14 Page 16 of 6 Y. Qiao, H. Li IEOT where we replace x with x 1 1 x x in (3.1) and substitute x z for x 1 y. Hence P x U g = U g P x1. This shows that P is invariant. Remark 4.1. For an invariant family P =(P x ), x (, ), if we take the limit as x, then we obtain that P is an integral operator with kernel p (x, y) = lim p(x x, x y). x For instance, if p(x, y) = a(x)f(xy 1 ), then P has integral kernel a()f (xy 1 ). Proposition We have π(p )= P,whereπ is the vector representation of Ψ ( H) on C c (, ). Proof. For all f C c (, ), we have ( Pf) r(x x, x 1 )= p(x x, y)f(y) dy y, and (π(p )f) r(x x, x 1 )) = P (f r(x x, x 1 )) = p(x x, x y)(f r φ x )(y) dy y = p(x x, x y)f(x y) dy y = p(x x, z)f(z) dz z This implies π(p )= P. Proposition There exists a unique invariant family P =(P x ), x (, ), so that π(p )= K, lim P x = K and lim P x x x = K, where K is defined in Eq. (5). Proof. We simply take p(t, s) tobek(t/s). Then the corresponding family defined above satisfies the requirements. We summarize what we have proved in the following theorem. Theorem The operator K can be (uniquely) identified with a smoothly invariant family P =(P x ), x [, ], on the groupoid H =[, ] (, ), such that π(p )= K, P = K, andp = K. Remark Since K is not uniformly supported, it does not belong to Ψ ( H) in the sense of [58]. However, it does belong to the pseudodifferental algebra of order on H constructed in [68]. However, since M r a KMr a is a smoothing operator (with smooth kernel) for a ( 1, 1), we obtain the following mapping property:

17 IEOT Double Layer Potentials on Polygons Page 17 of 6 41 Proposition For all k, l Z, anda ( 1, 1), we have M r a KMr a : H k (R +,g) H l (R +,g), where the metric g = r g e. Recall that H := [, ] (, ), where the action is given by dilation. We have Proposition If a ( 1, 1), then we have M r a KMr a C ( H). Proof. The kernel of M r a KMr a is ka (x, s) =k a (x/s) = 1 x 1+a s 1 a sin θ π x + s xs cos θ. Thus, it suffices to show that k a (x, s) belongs to L 1 ( H), that is, k a I <, where I is defined in Sect. 3. Indeed, we have k a I,d = 1 (xs 1 ) 1+a s 1 a sin θ ds sup π x [, ] (xs 1 ) + s (xs 1 )s cos θ s = 1 x 1+a s a sin θ ds sup π x [, ] x + s 4 xs cos θ s = 1 x 1+a e y(1 a) sin θ sup π x [, ] x + e 4y xe y cos θ dy = 1 (xe y ) a (xe y )sinθ sup π x [, ] x + e 4y xe y cos θ dy = 1 (xe y ) a sin θ sup π xe y + x 1 e y cosθ dy x [, ] = 1 e az sin θ sup π x [, ] (e z + e z cosθ) dz <. Hence, k a I,d is independent of x [, ] and it is finite if 1 <a<1. Similarly, we obtain k a I,r = 1 π = 1 π = 1 π = 1 π = 1 π <. sup x [, ] sup x [, ] sup x [, ] sup x [, ] sup x [, ] x 1+a s 1 a sin θ ds x + s xs cos θ s x 1+a e y(1 a) sin θ x + e y xe y cos θ dy (xe y ) a xe y sin θ x + e y xe y cos θ dy (xe y ) a sin θ xe y + x 1 e y cosθ dy e az sin θ e z + e z cosθ dz

18 14 Page 18 of 6 Y. Qiao, H. Li IEOT Thus, if 1 <a<1, then k a I,r is also finite and independent of x [, ]. As a consequence, k a I is finite, hence we have M r a KMr a C ( H). Recall that K θ is the double layer potential operator associated to the plane sector θ,andk θ is a matrix with diagonal and off diagonal K. Denote by P the pair groupoid of the set {1, }. According to the above discussion, we can identify K θ with a (smooth) invariant family of pseudodifferential operators on the Lie groupod H P which satisfies some requirements. Clearly, we can apply the same argument to M r ak θ M r a. Sowesummarize the results in the following proposition. Proposition Let a ( 1, 1). 1. There is a unique smooth invariant family Q =(Q (x,i) ), x [, ] and i {1, } on the Lie groupoid H P, such that π(q) =K θ,a, Q (,i) = K θ,a,andq (,i),wherei =1, and π is the vector representation.. We have M r ak θ M r a C ( H) M (C); 3. For all k, l Z, the following mapping property holds: M r ak θ M r a : K k 1 +a( θ) K l 1 +a( θ). 5. Double Layer Potentials on Plane Polygons Throughout this section, we use to denote a simply connected polygon in R with n successive vertices. We label these vertices as P 1,P,...,P n,p n+1 = P 1, the angle at vertex P i as θ i, and still denote by K the double layer potential operator associated to and the Laplace operator Δ. To get the Fredholmness property and invertibility of the operator ±I + K on some weighted Sobolev spaces on, we need some C -algebra knowledge and results in [11]. Motivated by the study of boundary value problem on (in the present paper, the domain is assumed not to have ramified cracks), Carvalho and Qiao associated to a (natural) Lie groupoid G [11]. Let us briefly review the construction in that paper. Denote by H =[, ) (, ) the action groupoid. For each angle θ i, we can associate the Lie groupoid J i = H P. Let M := \{P 1,P,...,P n } and M = M M be the pair groupoid of M. Then we can glue J i s and M in a certain way to obtain the Lie groupoid G. Let r be the (smoothened) distance function constructed in [7]. Recall that Ψ m (G) denotes the pseudodifferential operators of order m on G and by C (G) thec -algebra of the Lie groupoid G which is called layer potentials C -algebra in [11]. Proposition 5.1. If a ( 1/, 1/), then M r a KM r a C (G). Proof. Because the restriction of the double layer potential operator K (associated to ) to angle θ i, is just K θi which is a (smoothing) Mellin convolution operator discussed in Sect. 4, andsoism r a KM r a for a ( 1, 1). Hence, we can identify M r a KM r a with a unique smooth invariant family of operators

19 IEOT Double Layer Potentials on Polygons Page 19 of 6 41 on the Lie groupoid G such that the vector representation of the family at each angle is K θ,a (Sect. 4). According to the paper of Lewis and Parenti [35], the double layer potential operator K may be represented as an n n matrix [K i,j ] n i,j=1,and K i,j maps the functions on jth side to the function on ith side, and involves three possibilities: zero (i = j), K θi (if ith side and jth side do touch), and K i,j (if ith side and jth side do not touch). 1. If ith side and jth side are adjacent, then K i,j corresponds to K θi.thus, by Proposition 4.19, wehavem r a K θi M r a belongs to C (G) for a < 1.. If ith side and jth side do not touch, we need to consider the interaction among non-adjacent sides. In this case, since they are non-adjacent, the kernel of K i,j is bounded. However, we are using the metric g = r g e to identify weighted Sobolev spaces and usual Sobole spaces. As a consequence, to show that M r a K i,j M r a : K 1 ( ) K 1 ( ) is compact (hence belongs to the C -algebra C (G) ), it suffices to show that M r a K i,j M r a is a Hilbert Schmidt operator on L ( ), which requires that r a be square-integrable near, i.e., a<1/. By symmetry, the function r a should be square-integrable near as well, i.e., a> 1/. Hence, we obtain that a < 1/. Recall that by Proposition.3, we have the identification K m 1 ( ) H m (,g), where the metric g = r g e,andg e is the Euclidean metric. By the definition of weighted Sobolev spaces, we have K 1 ( ) r 1 L ( ). Let θ := min{ π π θ 1, π θ 1, π π θ, π θ,..., π π θ n, π θ n }. It is clear that 1/ <θ < 1. Proposition 5.. Let be a simply connected polygon on R,andK be the double layer potential operator associated to and the Laplace operator Δ. Then for a ( θ, 1/), the operators are both Fredholm. ±I + K : K 1 +a( ) K 1 +a( ) Proof. First of all, let us assume that a ( 1/, 1/). By Corollary 6.4 in [11] and Proposition 5.1, it is sufficient to prove that ±I + M r a KM r a is elliptic and the restriction of ±I + M r a KM r a to each angle θ i is invertible. The ellipticity of ±I + M r a KM r a is clear, and the invertibility of ±I + M r a KM r a (restricted to angle θ i )isprovedintheorem4.8. Therefore, we establish that for a ( 1/, 1/), are Fredholm. ±I + K : K 1 +a( ) K 1 +a( )

20 14 Page of 6 Y. Qiao, H. Li IEOT Secondly, we have the identification K( ) = L ( ). In [35], Lewis and Parenti showed that ±I + K : L ( ) L ( ) are isomorphisms. In particular, ±I + K are Fredholm operators. Hence the family of operators acting on L ( ) is still Fredholm for b <ɛfor some ɛ> small enough. In view of Proposition 4.1, we can take ɛ = θ 1/ >. Combining the above results, we obtain that for θ <a<1/, ±I +M r b KM r b are Fredholm. ±I + K : K 1 +a( ) K 1 +a( ) Lemma 5.3. Let be a polygon on R,andK be the double layer potential operator associated to and the Laplace operator Δ. The operators ±I + K : K 1 +a( ) K 1 +a( ) are injective for all a ( θ, 1/). Proof. In Proposition 4.1, we find all the possible singular values for double layer potentials at each vertex. The range of a in the lemma excludes all these values. Thus the conclusion holds for a ( θ, 1/). Theorem 5.4. Let be a simply connected polygon on R,andK be the double layer potential operator associated to and the Laplace operator Δ. The operators ±I + K : K m 1 +a( ) Km 1 +a( ) are both isomorphisms for all a ( θ, 1/). Proof. As in [7], the family of operators ±I + M r a KM r a : K 1 ( ) K 1 ( ) depends continuously on a. In[35], Lewis and Parenti already proved that ±I + K : L ( ) L ( ) are isomorphisms. By identification K ( ) = L ( ), we see that for a = 1 ( θ, 1/), ±I + K : K 1 +a( ) K 1 +a( ) are isomorphisms. Then by Proposition 5., we know that for all a ( θ, 1/), the operators ±I + K : K 1 +a( ) K 1 +a( ) are Fredholm of index zero. Moreover, since the operators M r a KM r a (regarded as a smooth invariant family of operators on the Lie groupoid G) area smooth family of operators with smooth kernels, the indices of the operators ±I + M r a KM r a : K m 1 ( ) K m 1 ( ) are independent of m. As a result, the operators ±I + M r a KM r a are all Fredholm of index zero. Then the desired result is followed by Lemma 5.3.

21 IEOT Double Layer Potentials on Polygons Page 1 of 6 41 Remark 5.5. For interior and exterior Neumann problems, we need to solve the boundary integral equations I + K and I + K (choosing a suitable fundamental solution of the Laplace operator), respectively, where K (x, y) = K(y, x) [1,35,69]. Thus, it is possible to extend our method to Neumann boundary value problems. 6. Conclusion In [11], to a plane polygon we associate a boundary groupoid G with the space of units given by a desingularization M of. The layer potentials C -algebra associated to is defined to be the groupoid convolution algebra C (G). In the present paper, we apply pseudodifferential operator (on Lie groupoids) techniques to the method of layer potentials to solve Dirichlet boundary value problems for the Laplace operator on a simply connected plane polygon. More precisely, let be a simply connected plane polygon with vertices P 1,P,...,P n. Denote by θ i the interior angle at vertex P i. The main ingredients of our proofs are as follows: 1. For each (infinite) plane sector θi, i =1,,...,n, since the double layer potential operator K θi (associated to θi and the Laplace operator Δ) is a Mellin convolution operator, we use the Mellin transform to prove that the operator K θi is invertible for suitable weighted Sobolev spaces on θi.. Using the properties of K θi and some results of Lewis and Parenti, we show that the double layer potential operator K (associated to and the Laplace operator Δ) belongs to the groupoid convolution algebra C (G). 3. Combining the invertibility of K θi and general results on pseudodifferential operators on Lie groupoids, we establish the Freholmness of the double layer potential operator K between appropriate weighted Sobolev spaces on. 4. We apply techniques from Melin transform to prove our main theorem. Namely, the operators ±I + K are in fact isomorphic between suitable weighted Sobolev spaces on. By the result in [7, Theroem 5.9], this implies a solvability result in weighted Sobolev spaces for the Dirichlet problem on. Acknowledgements We would like to thank Victor Nistor for helpful comments, enlightening discussions and communications. We both thank the anonymous referee who carefully read the paper and gave us useful suggestion on the range of weights. The first author would like to thank Wayne State University for the generous hospitality provided to him via the Fan China Exchange program by the American Mathematical Society.

22 14 Page of 6 Y. Qiao, H. Li IEOT References [1] Aastrup, J., Melo, S., Monthubert, B., Schrohe, E.: Boutet de Monvel s calculus and groupoids I. J. Noncommut. Geom. 4(3), (1) [] Ammann, B., Ionescu, A., Nistor, V.: Sobolev spaces on Lie manifolds and regularity for polyhedral domains. Doc. Math. 11, (6) (electronic) [3] Ammann, B., Lauter, R., Nistor, V.: Pseudodifferential operators on manifolds with a Lie structure at infinity. Ann. Math. () 165(3), (7) [4] Androulidakis, I., Skandalis, G.: The holonomy groupoid of a singular foliation. J. Reine Angew. Math. 66, 1 37 (9) [5] Androulidakis, I., Skandalis, G.: Pseudodifferential calculus on a singular foliation. J. Noncommut. Geom. 5(1), (11) [6] Angell, T.S., Kleinman, R.E., Král, J.: Layer potentials on boundaries with corners and edges. Časopis Pěst. Mat. 113(4), (1988) [7] Băcuţă, C., Mazzucato, A., Nistor, V., Zikatanov, L.: Interface and mixed boundary value problems on n-dimensional polyhedral domains. Doc. Math. 15, (1) [8] Cannas da Silva, A., Weinstein, A.: Geometric Models for Noncommutative Algebras. Volume 1 of Berkeley Mathematics Lecture Notes. American Mathematical Society, Providence (1999) [9] Carvalho,C., Nistor, V., Qiao, Y.: Fredholm conditions on non-compact manifolds: theory and examples. arxiv: [1] Carvalho, C., Nistor, V., Qiao, Y.: Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras. Electron. Res. Announc. Math. Sci. 4, (17) [11] Carvalho, C., Qiao, Y.: Layer potentials C -algebras of domains with conical points. Cent. Eur. J. Math. 11(1), 7 54 (13) [1] Costabel, M.: Boundary integral operators on curved polygons. Ann. Mat. Pura Appl. 4(133), (1983) [13] Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19(3), (1988) [14] Courant, R., Hilbert, D.: Methods of mathematical physics, vol. II. Wiley Classics Library. Wiley, New York (1989). Partial differential equations, Reprint of the 196 original, A Wiley-Interscience Publication [15] Debord, C., Skandalis, G.: Adiabatic groupoid, crossed product by R + and pseudodifferential calculus. Adv. Math. 57, (14) [16] Debord, C., Skandalis, G.: Pseudodifferential extensions and adiabatic deformation of smooth groupoid actions. Bull. Sci. Math. 139(7), (15) [17] Egorov, Y., Schulze, B.-W.: Pseudo-Differential Operators, Singularities, Applications. Volume 93 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel (1997) [18] Elschner, J.: The double layer potential operator over polyhedral domains. I. Solvability in weighted Sobolev spaces. Appl. Anal. 45(1 4), (199) [19] Fabes, E., Jodeit, M., Lewis, J.: Double layer potentials for domains with corners and edges. Indiana Univ. Math. J. 6(1), (1977) [] Fabes, E., Jodeit, M., Rivière, N.: Potential techniques for boundary value problems on C 1 -domains. Acta Math. 141(3 4), (1978)

23 IEOT Double Layer Potentials on Polygons Page 3 of 6 41 [1] Folland, G.: Introduction to Partial Differential Equations, nd edn. Princeton University Press, Princeton (1995) [] Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Volume 4 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston (1985) [3] Hsiao, G., Wendland, W.L.: Boundary Integral Equations. Volume 164 of Applied Mathematical Sciences. Springer, Berlin (8) [4] Jerison, D., Kenig, C.: The Dirichlet problem in nonsmooth domains. Ann. Math. () 113(), (1981) [5] Jerison, D., Kenig, C.: The Neumann problem on Lipschitz domains. Bull. Am. Math. Soc.(N.S.) 4(), 3 7 (1981) [6] Kapanadze, D., Schulze, B.-W.: Boundary-contact problems for domains with conical singularities. J. Differ. Equ. 17(), (5) [7] Kellogg, R.: Singularities in interface problems. In: Numerical Solution of Partial Differential Equations, II (SYNSPADE 197) (Proceedings of Symposia, University of Maryland, College Park, Md., 197), pp Academic Press, New York (1971) [8] Kenig, C.: Recent progress on boundary value problems on Lipschitz domains. In: Pseudodifferential Operators and Applications (Notre Dame, Ind., 1984). Volume 43 of Proceedings of Symposia in Pure Mathematics, pp American Mathematical Society, Providence (1985) [9] Kohr, M., Pintea, C., Wendland, W.: On mapping properties of layer potential operators for Brinkman equations on Lipschitz domains in Riemannian manifolds. Mathematica 5(75), (1) [3] Kondrat ev, V.: Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč. 16, 9 9 (1967) [31] Kozlov, V.A., Maz ya, V.G., Rossmann, J.: Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Volume 85 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1) [3] Kress, R.: Linear Integral Equations, nd edn. Volume 8 of Applied Mathematical Sciences. Springer, New York (1999) [33] Lauter, R., Monthubert, B., Nistor, V.: Pseudodifferential analysis on continuous family groupoids. Doc. Math. 5, (). (electronic) [34] Lauter, R., Nistor, V.: Analysis of geometric operators on open manifolds: a groupoid approach. In: Quantization of Singular Symplectic Quotients. Volume 198 of Progress in Mathematics, pp Birkhäuser, Basel (1) [35] Lewis, J., Parenti, C.: Pseudodifferential operators of Mellin type. Commun. Partial Differ. Equ. 8(5), (1983) [36] Li, H., Mazzucato, A., Nistor, V.: Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains. Electron. Trans. Numer. Anal. 37, (1) [37] Mackenzie, K.: Lie Groupoids and Lie Algebroids in Differential Geometry. Volume 14 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1987) [38] Maz ya, V.: Boundary integral equations. In: Analysis, IV. Volume 7 of Encyclopaedia of Mathematical Sciences, pp. 17. Springer, Berlin (1991)

24 14 Page 4 of 6 Y. Qiao, H. Li IEOT [39] Maz ya, V., Rossmann, J.: Elliptic Equations in Polyhedral Domains. Volume 16 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1) [4] Mazzeo, R., Melrose, R.: Pseudodifferential operators on manifolds with fibred boundaries. Asian J. Math. (4), (1998). Mikio Sato: a great Japanese mathematician of the twentieth century [41] Mazzucato, A., Nistor, V.: Well-posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks. Arch. Ration. Mech. Anal. 195(1), 5 73 (1) [4] McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge () [43] Medková, D.: The third boundary value problem in potential theory for domains with a piecewise smooth boundary. Czechoslov. Math. J. 47(4), (1997) [44] Melrose, R.: Transformation of boundary problems. Acta Math. 147(3 4), (1981) [45] Melrose, R.: The Atiyah Patodi Singer Index Theorem. Volume 4 of Research Notes in Mathematics. A K Peters Ltd., Wellesley (1993) [46] Melrose, R.: Geometric Scattering Theory, Stanford Lectures. Cambridge University Press, Cambridge (1995) [47] Mitrea, D., Mitrea, I.: On the Besov regularity of conformal maps and layer potentials on nonsmooth domains. J. Funct. Anal. 1(), (3) [48] Mitrea, I.: On the spectra of elastostatic and hydrostatic layer potentials on curvilinear polygons. J. Fourier Anal. Appl. 8(5), () [49] Mitrea, I., Mitrea, M.: The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains. Trans. Am. Math. Soc. 359(9), (7) (electronic) [5] Mitrea, M., Nistor, V.: Boundary value problems and layer potentials on manifolds with cylindrical ends. Czechoslov. Math. J. 57(4), (7) [51] Mitrea, M., Taylor, M.: Boundary layer methods for Lipschitz domains in Riemannian manifolds. J. Funct. Anal. 163(), (1999) [5] Mitrea, M., Taylor, M.: Potential theory on Lipschitz domains in Riemannian manifolds: L P Hardy, and Hölder space results. Commun. Anal. Geom. 9(), (1) [53] Moerdijk, I., Mrčun, J.: Introduction to Foliations and Lie Groupoids. Volume 91 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (3) [54] Monthubert, B.: Groupoids and pseudodifferential calculus on manifolds with corners. J. Funct. Anal. 199, (3) [55] Monthubert, B., Pierrot, F.: Indice analytique et groupoïdes de Lie. C. R. Acad. Sci. Paris Sér. I Math. 35(), (1997) [56] Nistor, V.: Pseudodifferential operators on non-compact manifolds and analysis on polyhedral domains. In: Booss, B., Grubb, G., Wojciechowski, K.P. (eds.) Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds. Volume 366 of Contemporary Mathematics, pp American Mathematical Society, Rhode Island (5)

25 IEOT Double Layer Potentials on Polygons Page 5 of 6 41 [57] Nistor, V.: Desingularization of Lie groupoids and pseudodifferential operators on singular spaces. [math.dg], to appear in Comm. Anal. Geom. [58] Nistor, V., Weinstein, A., Xu, P.: Pseudodifferential operators on differential groupoids. Pac. J. Math. 189(1), (1999) [59] Perfekt, K.-M., Putinar, M.: Spectral bounds for the Neumann Poincaré operator on planar domains with corners. J. Anal. Math. 14, (14) [6] Perfekt, K.-M., Putinar, M.: The essential spectrum of the Neumann Poincaré operator on a domain with corners. Arch. Ration. Mech. Anal. 3(), (17) [61] Qiao, Y., Nistor, V.: Single and double layer potentials on domains with conical points I: straight cones. Integral Equ. Oper. Theory 7(3), (1) [6] Renault, J.: A Groupoid Approach to C -Algebras. Volume 793 of Lecture Notes in Mathematics. Springer, Berlin (198) [63] Sauter, S., Schwab, C.: Boundary Element Methods. Volume 39 of Springer Series in Computational Mathematics. Springer, Berlin (11). Translated and expanded from the 4 German original [64] Schrohe, E., Schulze, B.-W.: Boundary value problems in Boutet de Monvel s algebra for manifolds with conical singularities. I. In: Pseudo-Differential Calculus and Mathematical Physics. Volume 5 of Mathematical Topics, pp Akademie Verlag, Berlin (1994) [65] Schrohe, E., Schulze, B.-W.: Boundary value problems in Boutet de Monvel s algebra for manifolds with conical singularities. II. In: Boundary Value Problems, Schrödinger Operators, Deformation Quantization. Volume 8 of Mathematical Topics, pp Akademie Verlag, Berlin (1995) [66] Schulze, B.-W.: Boundary value problems and singular pseudo-differential operators. Pure and Applied Mathematics (New York). Wiley, Chichester (1998) [67] Schulze, B.-W., Sternin, B. Shatalov, V.: Differential Equations on Singular Manifolds. Volume 15 of Mathematical Topics. Wiley-VCH Verlag Berlin GmbH, Berlin (1998). Semiclassical theory and operator algebras [68] So, B.K.: On the full calculus of pseudo-differential operators on boundary groupoids with polynomial growth. Adv. Math. 37, 1 3 (13) [69] Taylor, M.: Partial Differential Equations. II. Volume 116 of Applied Mathematical Sciences. Springer, New York (1996). Qualitative studies of linear equations [7] Verchota, G.: Layer potentials and regularity for the Dirichlet problem for Laplace s equation in Lipschitz domains. J. Funct. Anal. 59(3), (1984) [71] Verchota, G., Vogel, A.: The multidirectional Neumann problem in R 4.Math. Ann. 335(3), (6) Yu Qiao(B) School of Mathematics and Information Science Shaanxi Normal University Xi an 71119, Shaanxi China yqiao@snnu.edu.cn