On C. Neumann's Method for Second-Order Elliptic ... - Science Direct

Journal of Mathematical Analysis and Applications 262, 733᎐748 Ž2001. doi:10.1006rjmaa.2001.7615, available online at http:rrwww.idealibrary.com on

On C. Neumann’s Method for Second-Order Elliptic Systems in Domains with Non-smooth Boundaries O. Steinbach and W. L. Wendland Mathematisches Institut A, Uni¨ ersitat ¨ Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany E-mail: steinbach,[email protected] Submitted by William F. Ames Received April 17, 2001

DEDICATED TO GEORGE LEITMANN ON THE OCCASION OF HIS 75TH BIRTHDAY

In this paper we investigate the convergence of Carl Neumann’s method for the solution of Dirichlet or Neumann boundary values for second-order elliptic problems in domains with non-smooth boundaries. We prove that 12 I q K, where K is the double-layer potential, is a contraction in H 1r 2 Ž ⌫ . when an energy norm is used that is induced by the inverse of the single-layer potential. 䊚 2001 Academic Press

Key Words: boundary integral equations; Neumann series.

1. INTRODUCTION Carl Neumann’s famous method in potential theory goes back to C. F. Gauss w10x, who proposed to solve the Dirichlet problem for the Laplacian in a sufficiently smoothly bounded domain ⍀ by using a double-layer potential of the form

u Ž x . s Ž W¨ . Ž x . [

H⌫

ž

⭸ ⭸ ny

/

U* Ž x, y . ¨ Ž y . ds y

733 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.

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STEINBACH AND WENDLAND

with a double-layer charge ¨ to be determined as the solution of the boundary integral equation ¨s

ž

1 2

I q K ¨ y g , where Ž K¨ . Ž x . s

/

H⌫_ x4

ž

⭸ ⭸ ny

/

U* Ž x, y . ¨ Ž y . ds y .

U*Ž x, y . is the full-space fundamental solution of the Laplacian, and ⌫ [ ⭸ ⍀. Carl Neumann showed in his papers w26, 27x that, for a convex domain ⍀, the operator

Ž 12 I q K . is on the space of continuous functions strictly contracting with respect to the oscillation which, in fact, is equivalent to 5 12 I q K 5 - 1,

Ž 1.1.

with the operator norm generated by the norm 5 u 5 [ sup < u Ž x . y u Ž y . < q ␣ sup < u Ž x . < , x, yg⌫

Ž 1.2.

xg⌫

with an appropriate positive constant ␣ , on the Banach space C 0 Ž ⌫ . of continuous functions on ⌫ Žsee w16x.. In C. Neumann’s proof, completed in 1877 Žnot recognized in w16, 35x., only specific piecewise plane boundaries were excluded. For smooth Žincluding also non-convex. boundaries, J. Plemelj w29x showed for Ž 12 I q K . that its spectral radius on C 0 Ž ⌫ . is less than 1, and this result was extended by J. Radon to two-dimensional domains having boundaries of bounded rotation w30x; J. Kral finally found w18x necessary and sufficient conditions on the boundary curve. On L2 Ž ⌫ ., Shelepov w37x obtained a similar result under more restrictive assumptions. For three-dimensional domains and for smooth boundaries, e.g., a closed Lyapounov surface ⌫, Ž1.1. and Plemelj’s proof remain valid, as do C. Neumann’s result and Ž1.1., Ž1.2. for convex ⍀ w16x. For non-smooth and non-convex boundaries and the Laplacian, one finds sufficient but rather restrictive conditions on the geometry of ⌫ for Ž1.1. to hold with respect to the essential norm associated with C 0 Ž ⌫ . Ži.e., K is taken modulo compact perturbations. in the work by J. Kral w17x and by V. Maz’ya et al. in w1x Žfor the current state see Maz’ya’s survey in w22x. and in w41x. J. Kral and D. Medkova showed w23x that the essential Fredholm radius defined by I. Gohberg and A. Marcus w11x is here equivalent to finding an appropriate weighted supremum norm on C 0 Ž ⌫ .. For arbitrary polyhedral domains and the Laplacian, O. Hansen shows w13x how to construct the appropriate weight function. If L2 Ž ⌫ . is used then again Ž1.1.

SECOND-ORDER ELLIPTIC SYSTEMS

735

is valid in the spectral norm, as has been shown by J. Elschner w8x for a restricted class of boundaries ⌫. Carl Neumann’s iteration still converges in one of the Banach spaces on ⌫ if the corresponding spectral radius of the operator Ž 12 I q K . is less than 1. This property not only holds for the double-layer potential of the Laplacian but also for the double-layer potential of the Lame ´ system governing linearized elasticity if ⌫ is sufficiently regular. For a Lyapunov boundary ⌫ and the space of Holder continuous boundary functions these ¨ estimates for the spectrum of Ž 12 I q K . for the Lame ´ system go back to Mikhlin w24x and have further been elaborated in w19, 28, 32x. For boundary functions in L p , corresponding results are due to Shelepov w37x for two-dimensional problems and, for three-dimensional problems, are due to Maz’ya w21x and B. Dahlberg w6x. For a Lipschitz boundary ⌫ and the Laplacian, G. Verchota in w39x and, with the Lame ´ system, B. Dahlberg, C. Kenig, and G. Verchota in w7x showed that 12 I y K has closed range and is invertible on L2 Ž ⌫ .. However, these results do not provide that the spectral radius is less than 1. In this paper we employ coerciveness properties of the single layer and the hypersingular boundary integral operators which are still valid on a Lipschitz boundary ⌫ and which can be shown for the rather big class of second-order formally positive elliptic systems of partial differential equations w25, 34, 40x, based on results by M. Costabel w3x. Then the classical Neumann boundary integral equations of the second kind need to be considered in the appropriate boundary trace spaces H 1r2 Ž ⌫ . and Hy1 r2 Ž ⌫ ., respectively. Since the Calderon projection holds even on a Lipschitz boundary ⌫, we can show the strict contraction properties for Ž 12 I " K . and for the adjoint operators on appropriate subspaces of these trace spaces by using appropriate norms. Consequently, Carl Neumann’s classical iteration method, originally invented for solving the Dirichlet and Neumann problems of the Laplacian, converges for an astonishingly large class of problems. The applications are numerous, e.g., in domain decomposition with boundary elements as in w15x and for preconditioning as in w38x. The paper is organized as follows. In Section 2 we introduce the boundary integral operators of the first and second kind and appropriate norms on the trace spaces on ⌫. In Section 3 we formulate Carl Neumann’s method and the main results of the paper concerning the strict contracting properties. In Section 4 we comment on the so-called direct approach for boundary integral equations and apply appropriate Neumann series. In Section 5 we present the proofs of the main results Žwhich are formulated in Section 3.. The authors gratefully acknowledge the substantial support by the collaborative research centre ‘‘Sonderforschungsbereich 404, Mehrfeld-

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STEINBACH AND WENDLAND

probleme’’ of the German Research Foundation Deutsche Forschungsgemeinschaft at the University of Stuttgart. The second author also expresses his sincere gratitude to the Mittag᎐Leffler Institute of the Royal Academy of Sciences, Sweden, where he carried out some part of this work.

2. BOUNDARY INTEGRAL OPERATORS Let ⍀ ; ⺢ n Ž n s 2, 3. be a bounded domain with Lipschitz continuous boundary ⌫ s ⭸ ⍀. We consider a second-order, self-adjoint, formally positive elliptic partial differential operator L w25, 34, 40x given by

⭸

n

Lu Ž x . s y

Ý i , js1

⭸ xj

a ji Ž x .

⭸ ⭸ xi

⭸

n

uŽ x . q

Ý bi Ž x . ⭸ x u Ž x . q c Ž x . u Ž x . i

is1

Ž 2.1. and the corresponding conormal derivative operator Tx defined as n

Tx u Ž x . s

Ý

n j Ž x . a ji Ž x .

i , js1

⭸ ⭸ xi

uŽ x . ,

Ž 2.2.

where nŽ x . is the outer normal vector defined for almost every x g ⌫. Then there exists a fundamental solution U*Ž x, y . of L. The single-layer potential in the domain ⍀ and ⍀ c, respectively, is defined by

Ž Vw . Ž x . s H U* Ž x, y . w Ž y . ds y

for x g ⍀ j ⍀ c .

⌫

Ž 2.3.

For x g ⌫ we define the standard boundary integral operators, in particular, the single-layer potential,

Ž Vw . Ž x . s lim

H U* Ž x⬘, y . w Ž y . ds

x⬘ªx ⌫

y

for almost every x g ⌫, Ž 2.4.

where x⬘ g C x Ž ⍀ . and where C x Ž ⍀ . ; ⍀ denotes any cone with vertex x g ⌫ associated with the Lipschitz boundary ⌫ since it has the uniform cone property Žsee, e.g., w12, Theorem 1.2.2.2x.. In Ž2.4. we also let x⬘ g C x Ž ⍀ c .. We will also consider the double-layer potential, i

Ž W¨ . Ž x . [ H Ž Ty U* Ž x, y . . ¨ Ž y . ds y ⌫

for x g ⍀ j ⍀ c .

For appropriate ¨ , the corresponding double-layer boundary integral operator is defined in terms of its boundary values, namely by

Ž Ku . Ž x . [ lim Ž Wu . Ž x⬘ . q 12 u Ž x . x⬘ªx

for x g ⌫ and x⬘ g C x Ž ⍀ . .

Ž 2.5.

SECOND-ORDER ELLIPTIC SYSTEMS

737

Correspondingly,

Ž Ku . Ž x . [ lim Ž Wu . Ž x⬙ . y 12 u Ž x .

if x⬙ g C x Ž ⍀ c . .

x⬙ ªx

If ⌫ is locally Lyapunov in the vicinity of x g ⌫ then

Ž Ku . Ž x . s H

⌫_  x

Ž Ty U* Ž x, y . . 4

i

u Ž y . ds y ,

Ž 2.6.

where the integral is defined as a Cauchy principal value integral. ŽIf the principal part of ⌫ is the Laplacian then the kernel function of the integral operator in Ž2.6. is only weakly singular.. If x is a vertex point of ⌫ and the tangential cone to ⌫ at x approximates ⌫ in the vicinity of x ‘‘well enough,’’ then, for continuous u,

Ž Ku . Ž x . s ␴ Ž x . y

1 2

uŽ x . q

H⌫_ x4 Ž T U* Ž x, y . . y

i

u Ž y . ds y ,

and ␴ Ž x . is given explicitly Žsee, e.g., w41x for the Laplacian and w9x for the Lame ´ system.. For a Lipschitz boundary, Ž2.6. holds for almost every x g ⌫ only. The operator adjoint to K with respect to the duality ² ¨ , w :L 2 Ž⌫ . [

H⌫¨ Ž y . ⭈ w Ž y . ds , y

on appropriate pairs of function spaces on ⌫, is given in terms of the boundary values of the conormal derivative of the single-layer potential Ž2.3., namely by

⭸

n

Ž K ⬘w . Ž x . [ lim

Ý

x⬘ªx i , js1

n j Ž x . a ji Ž x .

⭸ xXi

Ž Vw . Ž x⬘ . y 12 w Ž x . , Ž 2.7.

where x⬘ g C x Ž ⍀ .. Again, if ⌫ is locally Lyapunov in the vicinity of x g ⌫, then

Ž K ⬘w . Ž x . s H

Tx U* Ž x, y . w Ž y . ds y ,

Ž 2.8.

⌫_  x 4

and the integral is defined as a Cauchy principal value integral. For a Lipschitz boundary, Ž2.8. holds for almost every x g ⌫. The hypersingular operator is defined by n

Ž Du . Ž x . [ yTx Ž Wu . Ž x . s y lim

Ý

x⬘ªx i , js1

n j Ž x . a ji Ž x .

⭸ ⭸ xXi

Ž Wu . Ž x⬘ . Ž 2.9.

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STEINBACH AND WENDLAND

for x g ⌫, where x⬘ g C x Ž ⍀ . or x⬘ g C x Ž ⍀ c .. If ⌫ is locally Lyapunov in the vicinity of x g ⌫ and u is smooth enough, then

Ž Du . Ž x . s yp.f. H

i

Tx Ž Ty U* Ž x, y . . u Ž y . ds y

⌫_  x 4

Ž 2.10.

for x g ⌫, where now the integral is defined as Hadamard’s finite part integral with respect to the Euclidean distance < x y y < Žsee w36x.. All of these operators as defined above will be considered here on the Sobolev spaces H s Ž ⌫ . on ⌫ for < s < F 1. The mapping properties of these boundary integral operators are well known Žsee, for example, w3, 5x., in particular, the boundary integral operators V : Hy1 r2qs Ž ⌫ . ª H 1r2qs Ž ⌫ . , K : H 1r2qs Ž ⌫ . ª H 1r2qs Ž ⌫ . , K ⬘: Hy1 r2qs Ž ⌫ . ª Hy1 r2qs Ž ⌫ . , D: H 1r2qs Ž ⌫ . ª Hy1 r2qs Ž ⌫ . are bounded for < s < - 12 Žsee w3x.. For the Laplacian and for the Lame ´ system, one still has continuity of K and K ⬘ even in L2 Ž ⌫ . due to the results in w39x and w6x. Furthermore, we assume that the single-layer potential in Ž2.4. is Hy1 r2 Ž ⌫ . elliptic, i.e., there exists a constant c1V ) 0 such that ² Vw, w :L 2 Ž⌫ . G c1V ⭈ 5 w 5 2H y1 r2 Ž⌫ .

for all w g Hy1 r2 Ž ⌫ . . Ž 2.11.

Remark 2.1. Assumption Ž2.11. is known for a wide class of partial differential operators L Žsee w5x.. However, in the case n s 2, appropriate scaling conditions are needed Žsee w14x for the Laplacian and w4x for the Bilaplacian; then Airy’s stress function yields corresponding scaling in 2D elasticity .. We denote by R [  ¨ g H 1r2 Ž ⌫ . : ¨ s ¨˜N ⌫ , ¨˜ g H 1 Ž ⍀ . : L¨˜ s 0 in ⍀ , T¨˜ s 0 on ⌫ 4

Ž 2.12. the finite-dimensional solution space of the homogeneous Neumann boundary value problem. Note that D¨ s Ž 12 I q K . ¨ s 0

for all ¨ g R .

Ž 2.13.

Let r2 Hy1 Ž ⌫ . [  w g Hy1r2 Ž ⌫ . : ² w, ¨ :L 2 Ž⌫ . s 0 for all ¨ g R 4 , 0

H01r2 Ž ⌫ . [  u g H 1r2 Ž ⌫ . : ² u, ¨ :L 2 Ž⌫ . s 0 for all ¨ g R 4 .

SECOND-ORDER ELLIPTIC SYSTEMS

739

As for V, we assume that the hypersingular integral operator D in Ž2.9. is H01r2 Ž ⌫ .-elliptic, i.e., there exists a constant c1D ) 0 such that ² Du, u:L 2 Ž⌫ . G c1D ⭈ 5 u 5 2H 1r 2 Ž⌫ .

for all u g H01r2 Ž ⌫ . .

Ž 2.14.

Since the single-layer potential V: Hy1 r2 Ž ⌫ . ª H 1r2 Ž ⌫ . is positive definite and bounded, we may define on Hy1 r2 Ž ⌫ . an equivalent Sobolev norm induced by V, i.e., 5w5V [

'²Vw, w :

L 2 Ž⌫ .

for all w g Hy1 r2 Ž ⌫ . .

Ž 2.15.

Correspondingly, we may equip H 1r2 Ž ⌫ . with an equivalent norm induced by the inverse of the single-layer potential Vy1 : 5 u 5 Vy1 [

y1

'²V

u, u:L 2 Ž⌫ .

for all u g H 1r2 Ž ⌫ . .

Ž 2.16.

These norms will be used to investigate the convergence of the Neumann series to be introduced in the next section.

3. CARL NEUMANN’S CLASSICAL METHOD Due to C. F. Gauss w10x, for the solution of the Dirichlet boundary value problem Lu Ž x . s 0 for x g ⍀ ,

uŽ x . s g Ž x .

for x g ⌫,

Ž 3.1.

C. Neumann investigated in w26, 27x the double-layer potential ansatz uŽ ˜ x . s Ž W¨ . Ž ˜ x.

for ˜ xg⍀

Ž 3.2.

with an unknown density ¨ g H 1r2 Ž ⌫ . for its solution. With the boundary values of the double-layer potential W, given by Ž2.5., we get from Ž3.2. the boundary integral equation of the second kind, 1 2¨

Ž x . y Ž K¨ . Ž x . s yg Ž x .

for all x g ⌫,

Ž 3.3.

for the yet unknown dipole boundary charge ¨ . To solve this equation, C. Neumann in w26, 27x showed for continuous g and L the Laplacian that the series ¨ Ž x. s y

⬁

Ý ž Ž 12 I q K .

ls0

l

g Ž x.

/

Ž 3.4.

converges uniformly on ⌫ provided ⌫ is convex. ŽThis was also the birth of the well-known Neumann series.. We consider now Ž3.4. on the trace space H 1r2 Ž ⌫ . instead and can show the convergence of Ž3.4. under very weak

740

STEINBACH AND WENDLAND

conditions on ⌫ and for a rather large class of equations due to the following main result of our paper. THEOREM 3.1. Let L be a formally positi¨ e elliptic partial differential operator of second order with a fundamental solution U*Ž⭈, ⭈ . generating all of the boundary integral operators Ž2.4. ᎐ Ž2.9.. Let the operators V and D satisfy the coerci¨ eness estimates Ž2.11. and Ž2.14.. Then, 5 Ž 12 I q K . u 5 Vy1 F c K ⭈ 5 u 5 Vy 1

for all u g H 1r2 Ž ⌫ .

Ž 3.5.

and 5 Ž 12 q K ⬘ . w 5 V F c K ⭈ 5 w 5 V

for all w g Hy1 r2 Ž ⌫ .

Ž 3.6.

with the positi¨ e constant cK [

1 2

q

'

1 4

y c1V c1D - 1.

Ž 3.7.

Here, c1V is the ellipticity constant of the single-layer potential V in Ž2.11. and c1D is the ellipticity constant of the hypersingular integral operator D in Ž2.14.. For the Neumann boundary value problem, i.e., to find the solution u of Lu Ž x . s 0

for x g ⍀ ,

Tx u Ž x . s f Ž x .

for x g ⌫, Ž 3.8.

r2 Ž . with given f g Hy1 ⌫ , where u is unique only modulo R, C. Neumann 0 employed the single-layer potential ansatz

uŽ ˜ x . s Ž Vw . Ž ˜ x.

for ˜ x g ⍀.

Ž 3.9.

Due to Ž2.7., here the boundary condition implies the boundary integral equation 1 2

w Ž x . q Ž K ⬘w . Ž x . s f Ž x .

for x g ⌫,

Ž 3.10.

now for the yet unknown boundary charge w. Since K ⬘ is the operator adjoint to K, J. Radon investigated in w30, Eq. Ž3.10.x, for the Laplacian in the space of bounded variation C*, the space dual to the continuous r2 Ž . functions on ⌫. Here, we now consider Ž3.10. in Hy1 ⌫ , which is dual to 0 1r2 Ž . 2 ² : H0 ⌫ with respect to ⭈ , ⭈ L Ž⌫ . . Now, we may find the solution again in the form of the Neumann series wŽ x. s

⬁

Ý ž Ž 12 I y K ⬘ .

ls0 r2 Ž . in Hy1 ⌫. 0

l

f Ž x.

/

Ž 3.11.

SECOND-ORDER ELLIPTIC SYSTEMS

741

r2 Ž . LEMMA 3.1. The operator 12 I y K ⬘ maps Hy1 ⌫ into itself boundedly. 0 r2 Ž . Proof. Let w g Hy1 ⌫ ; Hy1r2 Ž ⌫ .. Then the image z [ Ž 12 I y 0 y1 r2 Ž ⌫ . and satisfies K ⬘. w belongs to H

² z, ¨ :L 2 Ž⌫ . s ² w, ¨ :L 2 Ž⌫ . y ² z, Ž 12 I q K . ¨ :L 2 Ž⌫ . s 0

for all ¨ g R

r2 Ž . Ž ⌫ .. due to w g Hy1 ⌫ and because of Ž2.13.. Hence, z g Hy1r2 0 0 r2 Ž . The following theorem ensures the convergence of Ž3.11. in Hy1 ⌫. 0

THEOREM 3.2.

Let L and U*Ž⭈, ⭈ . be as in Theorem 3.1. Then,

5 Ž 12 I y K ⬘ . w 5 V F c K ⭈ 5 w 5 V

r2 for all w g Hy1 Ž ⌫. 0

Ž 3.12.

with the positi¨ e constant c K gi¨ en by Ž3.7. and

Ž 1 y c K . ⭈ 5 u 5 Vy1 F 5 Ž 12 I y K . u 5 Vy 1 F c K ⭈ 5 u 5 Vy 1 for all u g H01r2 Ž ⌫ . .

Ž 3.13.

Hence, this theorem guarantees the convergence of the Neumann series r2 Ž . Ž3.11. in Hy1 ⌫ . The proof of Theorem 3.2 will be given in Section 5. 0 4. BOUNDARY INTEGRAL EQUATIONS OF THE DIRECT APPROACH In the previous section we considered boundary integral equations resulting from the classical single- or double-layer potential ansatz, respectively. Now, we consider corresponding boundary integral equations of the second kind arising from the so-called direct approach based on Green’s representation formula. In general, the solution of our second-order boundary value problems in the interior is given by uŽ ˜ x. s

H⌫ U* Ž ˜x, y . t Ž y . ds

y

y

H⌫ Ž T U* Ž ˜x, y . . y

i

u Ž y . ds y

for ˜ x g ⍀.

Ž 4.1. Here, t Ž x . s Tx uŽ x . for x g ⌫; and u N ⌫ and t are the Cauchy data of L in ⍀. When the Dirichlet boundary value problem Ž3.1. is considered, then the Neumann datum t Ž x . on ⌫ is unknown. Hence, for finding t we may consider either a boundary integral equation of the first kind Žsee, e.g., w14x., or a boundary integral equation of the second kind, i.e.,

Ž 12 I y K ⬘ . t Ž x . s Ž Dg . Ž x .

for x g ⌫.

Ž 4.2.

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STEINBACH AND WENDLAND

Again, we may use a Neumann series to compute t, the solution of Ž4.2., i.e., tŽ x. s

⬁

Ý ž Ž 12 I q K ⬘ .

l

for x g ⌫.

Dg Ž x .

/

ls0

Ž 4.3.

The second assertion Ž3.6. of Theorem 3.1 now ensures the convergence of the Neumann series Ž4.3. in the space Hy1 r2 Ž ⌫ .. In the case of the Neumann boundary value problem Ž3.8., the Dirichlet datum uŽ x . on ⌫ is unknown. Hence, the solution of the boundary integral equation

Ž 12 I q K . u Ž x . s Ž Vg . Ž x .

for x g ⌫

Ž 4.4.

will provide us with the desired boundary values of u. Note that u g H 1r2 Ž ⌫ . is unique only modulo R since Ž4.1. implies that

Ž 12 I q K . u 0 s 0

for every u 0 g R .

We therefore define the projection operator P R : H the mapping u ¬ P R u via ² P R u, ¨ :L 2 Ž⌫ . s 0

1r2 Ž

Ž 4.5. ⌫. ª

H01r2 Ž ⌫ .

for all ¨ g R .

by

Ž 4.6.

Instead of Ž4.4., now we consider the uniquely solvable boundary integral equation I y P R Ž 12 I y K . u s P R Vg ,

Ž 4.7.

to find u g H01r2 Ž ⌫ .. The Neumann series for the solution of Ž4.7. is given by uŽ x . s

⬁

Ý

l

P R Ž 12 I y K . P R Ž Vg . Ž x .

for x g ⌫.

Ž 4.8.

ls0

Because of Assertion Ž3.13., this Neumann series converges in H01r2 Ž ⌫ .. 5. PROOFS The proofs are based on the properties of the Calderon projection operator associated with L in ⍀. We write both boundary integral equations as a system resulting from the direct approach in the form of the Calderon projection, i.e., u Tx u s

ž /

ž

1 2

IyK D

V 1 2

I q K⬘

u Tx u .

/ž /

Ž 5.1.

SECOND-ORDER ELLIPTIC SYSTEMS

743

From Ž5.1. we derive the Dirichlet᎐Neumann map for x g ⌫ Ž Tu . Ž x . [ Ž Su . Ž x . in the form of the Steklov᎐Poincare ´ operator S given by

Ž 5.2.

S s Vy1 Ž 12 I q K . s D q Ž 12 I q K ⬘ . Vy1 Ž 12 I q K . .

Ž 5.3.

Now we are in a position to formulate some results based on the projection property of the Calderon projection Ž5.1. as well as on the mapping properties of all of the boundary integral operators in Ž2.4. ᎐ Ž2.10.: PROPOSITION 5.1. The double-layer potential operator K as defined in Ž2.5., Ž2.6. can be symmetrized; in particular, KV s VK ⬘,

DK s K ⬘D

and

VD s 14 I y K 2 ,

DV s 14 I y K ⬘ 2 .

Ž 5.4. Proof. The relations in Ž5.4. are a direct consequence of the projection property of Ž5.1.. In the special case of the Laplacian in a smoothly bounded two-dimensional domain, this result was already given by J. Plemelj w29x and for boundaries of bounded rotation by J. Radon w30x. For the Laplacian and boundaries with bounded cyclic variations, the first relation in Ž5.4. was shown by J. Kral Žsee w17x.. PROPOSITION 5.2. holds

Let V: Hy1 r2 Ž ⌫ . ª H 1r2 Ž ⌫ . satisfy Ž2.11.. Then there

c1V ⭈ ² Vy1 u, u:L 2 Ž⌫ . F 5 u 5 2H 1r 2 Ž⌫ .

for all u g H 1r2 Ž ⌫ .

Ž 5.5.

with the same positi¨ e constant c1V as in Ž2.11.. Proof. For w g Hy1 r2 Ž ⌫ ., w / 0, by duality,