SPECTRAL PROPERTIES OF THE SIMPLE LAYER POTENTIAL TYPE OPERATORS

Transcription

1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 43, Number 3, 2013 SPECTRAL PROPERTIES OF THE SIMPLE LAYER POTENTIAL TYPE OPERATORS MILUTIN R. OSTANIĆ ABSTRACT. We establish the exact asymptotical behavior of singular values of the simple layer potential type operators. 1. Introduction notation. It is well known that any function f C 2 Ω Ω R n,n 2 may be expressed as ux y f x = u x y f y dy + f y ds y Ω Ω n y f u x y ds y x Ω, Ω n y where { 1/n 2 σn x n 2, n > 2 u x = 1/2π ln 1/ x, n = 2 σ n =2π n/2 /Γn/2 is the area of n 1-dimensional sphere. Here, Ω is a domain in R n, is the Laplace operator, / n y denotes the derivative in the direction of external normal to Ω with respect to y ds y denotes the area element. Operators g u x y g y dy, Ω g u x y g y ds y g Ω Ω ux y n y g y ds y 2010 AMS Mathematics subject classification. Primary 47B06, Secondary 31A10. Keywords phrases. Simple layer potential, asymptotics of singular values. Partially supported by MNZZS Grant No Received by the editors on January 22, 2009, in revised form on October 8, OI: /RMJ Copyright c 2013 Rocky Mountain Mathematics Consortium 855

2 856 MILUTIN R. OSTANIĆ are called, respectively, operators of volume potential, simple layer potential double layer potential type. Spectral properties of volume potential type more general operators on L 2 Ω have been investigated in a number of papers see for example [1 5, 7, 16]. We use the potentials of the simple layer double layer to solve boundary problems. We seek the solution of the inner irichlet problem in the form of a potential of double layer. Moreover, we seek the solution of the external irichlet problem in the form of a potential of double layer plus a constant. Finally, we seek the solution of the Neumann problem inner or external in the form of a potential of simple layer. In this way, solving the boundary problem is reduced to integral equations with polar kernels. As for a review of the classical methods of the potential simple double layer for harmonic function in domains with smooth boundary, see [9, 14]. In the paper [10], a general rigorous account of Poincare s variational problem which is related to solvability of irichlet problem is given using method of modern analysis. More precisely, the authors interpreted Poincare s variational principle as a nonselfadjoint eigenvalue problem for angle operator between two distinct pairs of subspaces of potentials. An example of use of the method of potentials applied to equations of the second order with nonconstant coefficients may be seen in [11]. To the author s knowledge, spectral properties of the simple layer potential type operators acting from L 2 Ω to L 2 Ω haven t been studied so far. In this paper, the exact asymptotic of the singular values of the simple layer potential type operators will be established for n =2,aswell as the connection with the length of the boundary Ω of domain Ω. For the rest of this paper, Ω will denote the length of the boundary Ω of domain Ω C, while L 2 Ω will denote the class of functions f such that Ω f 2 dz < dz is the arclength measure on Ω. Finally, da will denote Lebesgue measure in Ω.

3 SIMPLE LAYER POTENTIAL TYPE OPERATORS 857 If H 1 H 2 are Hilbert spaces T : H 1 H 2 is a compact operator, then T T : H 1 H 1 is compact, selfadjoint nonnegative T = T T is a unique nonnegative square root of operator T T. Eigenvalues of operator T, i.e., λ n T, are called the singular values of operator T. Hence, s n T =λ n T wheres n T denotes singular values of operator T. For properties of singular values in more detail see, for example, [8, 15]. Let N t S denote the singular values distribution function of operator S, i.e., N t S = 1 t>0. s ns t The symbol K Sx, y dμ will denote an integral operator on L2 K, μ, with kernel Sx, y. 2. Main result. Theorem 1. Let Ω be a bounded, simply connected domain in C with an analytic boundary, let be an operator defined by Tfz = 1 2π T : L 2 Ω L 2 Ω Ω ln z ξ f ξ dξ. Then 3/2 Ω s n T, n. 2πn Here, symbol a n b n denotes the fact that lim n a n /b n =1. Note that operator T assigns harmonic functions on C \ Ω to functions in space C Ω [14]. Furthermore, if f L 2 Ω, then ln z ξ f ξ da ξ C Ω. Ω

4 858 MILUTIN R. OSTANIĆ Remark 1. Eigenvalues eigenfunctions of the operator of the logarithmic potential on the unit disc were found in [1, 2]. Also, in the case of arbitrary bounded domain Ω C, two-sided estimates of the growth of singular values of that operator were given. In addition, asymptotic behavior of singular values of the product of the Bergman projection the operator of logarithmic potential was studied. An exact asymptotic of s n T, where T is the operator from Theorem 1, is given. The operator T acts from L 2 Ω into L 2 Ω the growth order of s n T is precisely between the growth order s singular values of logarithmic potential operator singular values of its product with Bergman projection. In the presence of corners the investigation of s n T, using our method, is more difficult because the derivative of conformal map ϕ : Ω is unit disc has singularities on Ω inthatcase. The presence of corner manifests perhaps in the higher-order terms of asymptotics of s n T. Note that the assumption of the analytic boundary can be relaxed; we choose to work with the analytic boundary to make exposition simpler. However, using Kellog s theorem, one can prove the same result for C k boundary k large enough. Several lemmas will be needed for the proof of Theorem 1. Lemma 1 [3]. Let H be a compact operator, r>0, let for any ε>0 there exist decomposition H = H ε + H ε where H ε H ε are compact operators for which the following holds: 1. lim n nr s n H ε=ch ε 2. lim n nr s n H ε ε. Then, there exists lim ε 0 + CH ε=ch, furthermore, lim n nr s n H =C H. Lemma 2. If H : L 2 0,a L 2 0,a is an operator defined by Hfx = a 0 x + y 2 ln x + y f y dy,

5 SIMPLE LAYER POTENTIAL TYPE OPERATORS 859 then lim n n3 s n H =0. Proof. Applying integration by parts, one obtain H = R 8 +2H 1 J 2 where R 8 is an operator on L 2 0,aofrank 8, H 1,J : L 2 0,a L 2 0,a H 1 f x = Jf x = a 0 x 0 ln x + y f y dy f y dy. Since [6], lim n ns n+1 H 1 = 0, since s n J const 1/n, from the property of singular values of the product of operators Ky-Fan theorem [8] it follows that lim n n3 s n H =0. Lemma 3. If p C[ π, π], px > 0 on [ π, π], then for operator : L 2 [ π, π] L 2 [ π, π] defined by f x = π π p x p yx y 2 ln x y f y dy, the following holds: s n 2 π 3 π 2 n 3 p x dx 2/3, n. π Proof. Let us first demonstrate that 1 1 s n x y 2 ln x y dy 16 n 3 π 2. 1

6 860 MILUTIN R. OSTANIĆ It is known [13] that 2 e it ξ π K 0 ξ dξ =. 1+t 2 R Here K 0 z = I 0 zlnz/2 + k=0 [z/22k /k! 2 ] Ψk +1 is a Mconald function, I 0 z = k=0 [z/22k /k! 2 ], Ψn+1 = n k=1 1 k γ, γ is an Euler constant, Ψ1 = γ. From 2, it follows that e it ξ ξ 2 K 0 ξ dξ = d2 dt 2 π 1+t 2 1/2 2π,t +. t3 Applying [7, Theorem 1] to integral operator 1 kx y dy with kernel 1 kt =t 2 K 0 t, Lemma 2 the Ky-Fan theorem, one obtains 1. Theorem 1 from [7] saysthatifk is an even function which decreases rapidly at Kξ = R eitξ kt dt is a decreasing function with the asymptotic Kξ Lξ/ξ r, ξ + r N, L is a slowly varying function if s n 2 0 k x + y dy = o L n n r, then s n 1 1 k x y dy L n nπ/2 r, n. From 1, it follows that 3 s n x y 2 ln x y dy 2 3 n 3 π 2, n, where is an interval its length. Let i =[ π+2π/ni 1, π+2π/ni], ξ i i, i =1, 2,...,N. Let P i 1 i N denote linear operators on L 2 [ π, π] defined by P i f x =X i x f x X S -characteristic function of set S.

7 SIMPLE LAYER POTENTIAL TYPE OPERATORS 861 Furthermore, let i : L 2 [ π, π] L 2 [ π, π] 1 i N denote operators defined by Then Obviously, i f x = π π p ξ i 2 x y 2 ln x y f y dy. N N = P i i P i + P i i P i + s 2n P i i P i N P i P j. i j i,j=1 = s 2n p xpy p ξ i x y 2 ln x y dy i + p ξ i px p ξ i x y 2 ln x y dy. i Let ε>0. Choose N large enough such that px pξ i <εfor all x i all 1 i N. Keeping this in mind, one obtains 4 s 2n P i i P i 2 ε p s n x y 2 ln x y dy. i From 3 4, it follows that s 2n P i i P i 2 ε p C i 3 where constant C does not depend on n nor on i {1, 2,...,N}. From the last inequality, it follows that 5 s n P i i P i C i 3 ε where the constant C does not depend on n nor on i {1, 2,...,N}. n 3 n 3

8 862 MILUTIN R. OSTANIĆ Since operators P i i P i 1 i N are orthogonal, it follows that N N 6 N t P i i P i = N t P i i P i. From 5 it follows that N t P i i P i so, from 6, one obtains N N t P i i P i 1/3 Cε i, i =1, 2,...,N, t 1/3 Cε N i =2π t 1/3 Cε. Putting in this inequality t = s n N P i i P i, one obtains N 7 s n P i i P i 8π3 Cε n 3. According to Lemma 2, it follows that for all i, j {1, 2,...,N} such that i j = 1, the following holds 8 lim n n3 s n P i P j =0. Indeed, if for example i = j +1, then the singular values of the operator P i P i+1 are equal to the singular values of the operator Δ i p x p yx y 2 ln x y dy : L 2 Δ i L 2 Δ i+1. A change of variables x =2π/Ni x 1, y =2π/Ni + y 1, x 1,y 1 [0, 2π/N] keeping in mind that the mappings f f2π/ni x are isometries between L 2 Δ i L 2 Δ i+1 L 2 0, 2π/N reduces the previous operator to the form for application of Lemma 2. If i j 2, then singular values of operators P i P j have an exponential decrease Birman-Solomyak theorem [4, Theorem 11.3, t

9 SIMPLE LAYER POTENTIAL TYPE OPERATORS 863 page 78] so, from 8 based on properties of singular values of the sum of operators, one obtains N 9 lim n n3 s n P i P j =0. i j i,j=1 Let N N E N = P i i P i + P i P j. Then, from 7 8, it follows that i j 10 lim n n3 s n E N 8π 3 C ε. Let N E N = P i i P i Operators P i i P i 1 i N are mutually orthogonal so From 3, it follows that so we obtain N N t E N = N t P i i P i. 2p ξi 2 i 3 1/3 N t P i i P i π 2, t 0 + t N 11 lim t 1/3 N t E t 0 + N = 1/3 2 π 2 p ξ i 2/3 def i = ω N. Putting t = s n E N in 11, one obtains 12 lim n n3 s n E N =ω3 N.

10 864 MILUTIN R. OSTANIĆ Since = E N + E N, from applying Lemma 1, we obtain lim n n3 s n = lim N ω3 N = 2 π 3 π 2 p x dx 2/3, π which proves Lemma 3. Lemma 4. Let k 0 t = 1 e int + e int 16π n 2 n +1, n 1 p C[ π, π], px > 0 on [ π, π] W 0 : L 2 π, π L 2 π, π W 0 f x = π π p x p y k o x y f y dy. Then s n W 0 1 π 3 p x 2/3 1 dx 2π π n 3, n. Proof. Since ln 2sin t = 2 this follows from the expansion n=1 ln 1 re i x = cos nt, t <π, t 0, n n=1 r n ein x n taking real part from both sides putting r 1, transforming function k 0 for t <π, t 0weget k 0 t = 1 [ 1 8π cos t ] cos nt n 2 n 2 1 n=2

11 SIMPLE LAYER POTENTIAL TYPE OPERATORS 865 [ ] 1 cos t 2 sin t /2 + ln 8π t π t /2 [ ] 1 cos t t2 2 + ln t 8π + 1 [ 1 cos tln π t ] 8π 2 [ ] π t2 ln t 5 = r i t. Let M p,w 1,W 2,W 3,W 4,W 5 be linear operators on L 2 π, π defined by Then M p f x =p x f x W i = π π 13 W 0 = r i x y dy i =1, 2, 3, 4, 5. 5 M p W i M p. It is easily checked that 1 14 s n W 1 =O From the Birman-Solomyak theorem [4, Theorem 11.3, page 78] on asymptoticness of singular values of operators with analytic kernel it follows that n s n W 2 =O e δn, δ>0 does not depend on n.

12 866 MILUTIN R. OSTANIĆ From [8, page 157] it follows that 1 16 s n W 3 =O from Lemma 2, it follows that n 7/2 17 s n W 4 =o n 3. Indeed, if P +,P : L 2 π, π L 2 π, π are operators defined by P + f = X [0,π] f, P f = X [ π,0] f, thenwehave, W 4 = P + W 4 P + + P W 4 P + + P + W 4 P + P W 4 P. For operators P + W 4 P +, P W 4 P we have according to [8, pages ] s n P + W 4 P + =o n p s n P W 4 P =o n p for every p N. The isometry f f x between L 2 0,πL 2 π, 0 reduce operators P W 4 P +, P + W 4 P to the suitable form for application to Lemma 2 so, s n P + W 4 P =o n 3 s n P W 4 P + =o n 3. Since, according to Lemma 3, 1 π 3 s n M p W 5 M p 2nπ 3 p x dx 2/3, n, π since operator M p theorem, it follows that is bounded, from the Ky-Fan s n W 0 1 π 3 p x 2/3 1 dx 2π π n 3, n. Let ϕ be the conformal mapping of = {z : z < 1} onto Ω. According to Schwarz s theorem, ϕ can be analytically extended to the neighborhood, ϕ z 0on.

13 SIMPLE LAYER POTENTIAL TYPE OPERATORS 867 Let A 3 z,ξ = ϕ z ϕ ξ 4π 2 ln t z ln t ξ da t G : L 2 L 2 be an operator defined by Gf z = ϕ z ϕ ξa 3 z,ξ f ξ dξ. Lemma 5. The following asymptotic holds: 3 Ω s n G, n. 2πn Proof. Let Then Ω 0 z,ξ = 1 4π 2 ln t z ln t ξ da t G 0 z,ξ=ϕ z 3/2 ϕ ξ 3/2 Ω 0 z,ξ. Gf z = From this, it follows that 18 V 2 G = WV 1, G 0 z,ξ f ξ dξ. where V 1, V 2 : L 2 π, π L 2 π, π are isometries defined by V 1 fx =f e ix ϕ e ix 3/2 ϕ e ix 3/2 V 2 fx =f e ix ϕ e ix 3/2 ϕ e ix 3/2,

14 868 MILUTIN R. OSTANIĆ W : L 2 π, π L 2 π, π is an operator defined by Wf x = π π p x p yω 0 e ix,e iy f y dy where px = ϕ e ix 3/2. From 18 it follows that s n G =s n W so, keeping in mind that Ω 0 e ix,e iy = 1 4π 2 ln t e ix ln t e iy da t = k 0 x y since ln t e ix 1 = 2 ln 1 te ix ln 1 te ix = 1 1 t n e inx + t n e inx 2 n ln t e iy = 1 2 n=1 n=1 1 t n e iny + t n e iny, t < 1, n then multiplying integrating with respect to dat over, we obtain previous equality, applying Lemma 4 substituting px = ϕ e ix 3/2 in it, one obtains s n G which proves Lemma 5. 1 π ϕ e ix 3 1 dx 2π π n 3 = 3 Ω, n, 2πn 3. Proof of Theorem 1. It is easily established that operator T : L 2 Ω L 2 Ω acts in the following way T f z = 1 ln ξ z f ξ da ξ, 2π so T T : L 2 Ω L 2 Ω is an operator defined by T Tfz = K z,ξ f ξ dξ Ω Ω

15 SIMPLE LAYER POTENTIAL TYPE OPERATORS 869 where K z,ξ = 1 4π 2 ln t z ln t ξ da t. Ω Let V : L 2 Ω L 2 S 1 : L 2 L 2 be operators defined by Vfz =f ϕ z ϕ z S 1 f z = Since V is an isometry ϕ z ϕ ξk ϕ z,ϕξ f ξ dξ. VT T = S 1 V, one obtains i.e., s n T T =s n S 1, 19 s 2 n T =s n S 1. Let M 1,S 2 : L 2 L 2 be operators defined by S 2 f z = K ϕ z,ϕξ f ξ dξ M 1 f z = ϕ zf z. Operator M 1 is bounded because ϕ is analytic function on neighborhood. Then we have M1 ϕ f z = zf z 20 S 1 = M 1 S 2 M1.

16 870 MILUTIN R. OSTANIĆ The kernel of operator S 2, i.e., the function Kϕz,ϕξ may be exped in the following way: K ϕ z,ϕξ = 1 4π 2 ln ϕ t ϕ z t z ln ϕ t ϕ ξ t ξ ϕ t 2 da t ln t z ln ϕ t ϕ ξ t ξ ϕ t 2 da t ln ϕ t ϕ z t z ln t ξ ϕ t 2 da t ln t z ln t ξ ϕ t 2 da t + 1 4π π π 2 Therefore, S 2 = 4 are defined by = K 1 z,ξ+k 2 z,ξ+k 3 z,ξ+k 4 z,ξ. Si 2 where operators S i 2 : L 2 L 2 S i 2 f z = K i z,ξ f ξ dξ. Let us demonstrate that operators S i 2 1 i 3 have an exponential decrease of singular values. Note that S 1 2 = PQ where P : L 2 L 2, Q : L 2 L 2 Pf z = 1 4π 2 Qf z = 1 4π 2 ln ϕ t ϕ z t z ϕ t 2 f t da t ln ϕ t ϕ z t z f t dt. Operator P is bounded, while operator Q according to the Birman- Solomyak theorem [4, Theorem 11.3, page 78] has exponential decay of singular values. Therefore, singular values of operator S 1 2 have exponential decay. In a similar way it may be demonstrated that singular values of operator S 2 2 S 3 2 also have exponential decay.

17 SIMPLE LAYER POTENTIAL TYPE OPERATORS 871 Kernel K 4 may be expressed as K 4 z,ξ = 1 4π 2 ϕ t ϕ z ln t z ln t ξ ϕ t dat + 1 4π 2 ϕ z ln t z ln t ξ ϕ t ϕ ξ da t + ϕ z ϕ ξ 4π 2 ln t z ln t ξ da t = A 1 z,ξ+a 2 z,ξ+a 3 z,ξ. Then S 4 2 = A 1 + A 2 + A 3 where A i : L 2 L 2 A i f z = A i z,ξ f ξ dξ 1 i 3. Let us demonstrate that s n A 1 =o n 3. Note that where A 1 = P 1 M ϕ Q 1, P 1 : L 2 L 2 Q 1 : L 2 L 2 M ϕ : L 2 L 2 P 1 f z = 1 4π 2 ϕ t ϕ z ln t z f t da t Q 1 f z = ln t z f t dt M ϕ f z =ϕ z f z.

18 872 MILUTIN R. OSTANIĆ Let H 1,H 2 : L 2 L 2 H 1 f z = t z ln t z f t da t H 2 f z = [ϕ t ϕ z ϕ zt z] f t da t Clearly, P 1 = 1 4π 2 M ϕ H 1 + H 2 M ϕ -operator of multiplying by ϕ on L 2. The singular values of the operator H 1 can be directly calculated, we obtain s n H 1 =On 5/2 so s n M ϕ H 1 =O n 5/2. From the Paraska theorem [12] it follows that s n H 2 =o n 3/2 which, together with the previous formula, gives 21 s n P 1 =o n 3/2. The singular values of operator Q 1 can also be directly calculated, we get s n Q 1 =O n 3/2. Keeping in mind that M ϕ is a bounded operator that s n Q 1 = On 3/2 is easily obtained from 21 the fact that A 1 = P 1 M ϕ Q 1, it follows that 22 s n A 1 =o n 3. In a similar way, one demonstrates 23 s n A 2 =o n 3.

19 SIMPLE LAYER POTENTIAL TYPE OPERATORS 873 Let R = S S S A 1 + A 2. Since operator S S S 3 2 has exponential decay of singular values, from 22 23, it follows that 24 s n R =o n 3. Since S 2 = R + A 3,wehave S 1 = M 1 S 2 M 1 = M 1RM 1 + M 1A 3 M 1. Since M 1 A 3 M 1 = G operator from Lemma 5 we get S 1 = G + M 1 RM 1. Keeping in mind Lemma 5, as well as the fact that M 1 is a bounded operator, from 24, the Ky-Fan theorem the previous equality, it follows that 3 Ω s n S 1, n 2πn so, from 19, we obtain s n T which proves Theorem 1. 3/2 Ω, n, 2πn Theorem 2. For the operators ln z ξ dξ Ω Ω ξ z α 1 dξ,

20 874 MILUTIN R. OSTANIĆ which act on L 2 Ω, the following asymptotic formula holds: s n Ω ln z ξ dξ Ω n, n s n Ω ξ z α 1 dξ 2Γα cos απ Ω α n α, 2 α >0, α 1, 3, 5,.... n Remark 2. The proof of Theorem 2 is similar to the proof of Theorem 1, but simpler. Using the conformal map ϕ : Ω, study of asymptotics of singular values of the operator from Theorem 2 is reduced to the investigation of spectral properties of integral operators on L 2 0, 2π with kernels of the form axbyln x y or axby x y α 1. The operator Ω ln z ξ dξ, from Theorem 2, acts on space L2 Ω while the operator from Theorem 1 acts from L 2 Ω to L 2 Ω, this is a reason for different asymptotics of their singular values. Acknowledgments. The author is grateful to the referee for the suggestion which helped to improve this article. REFERENCES 1. J.M. Anderson,. Khavinson V. Lomonosov, Spectral properties of some integral operators arising in potential theory, Quart.J.Math.Oxford , J. Arazy. Khavinson, Spectral estimates of Cauchy s transform in L 2 Ω, Int. Eq. Oper. Theor , M.Š. Birman M.Z. Solomjak, Asymptotic behavior of the spectrum of weakly polar integral operators, Izv. Akad. Nauk , , Estimates of singular values of the integral operators, Uspek. Mat. Nauk , M.R. ostanić, Spectral properties of the Cauchy operator its product with Bergman s projection on a bounded domain, Proc. Lond. Math. Soc ,

21 SIMPLE LAYER POTENTIAL TYPE OPERATORS M.R. ostanić, Asymptotic behavior of the singular values fractional integral operators, J. Math. Anal. Appl , , Exact asymptotic behavior of the singular values of integral operators with the kernel having singularity on the diagonal, Publ. L Institut Math , I.C. Gohberg M.G. Krein, Introduction to the theory of linear nonselfadjoint operators, American Mathematical Society, Providence, R.I., O.. Kellog, Foundations of potential theory, Springer, Berlin, Khavinson, M. Putinar H. Shapiro, On Poincare s variational problem in potential theory, Arch. Rat. Mech. Anal , C. Mira, Partial differential equations of elliptic type, Second edition, Springer-Verlag, Berlin, V.I. Paraska, On asymptotics of eigenvalues singular numbers of linear operators which increase smoothness, Math. SB , in Russian. 13. S.G. Samko, A.A. Kilbas O.I. Maricev, Fractional integrals derivations some applications, Minsk, 1987 in Russian. 14. V.S. Vladimirov, Equations of mathematical physics, Moscow, 1981 in Russian. 15. J. Weidmann, Linear operators in Hilbert space, Springer-Verlag, New York, H. Widom, Asymptotic behavior of the eigenvalues of certain integral equations, Trans. Amer. Math. Soc , University of Belgrade, Faculty of Mathematics, Studentski trg 16, Belgrade, Serbia address: domi@matf.bg.ac.rs