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Download On PoinCare's Variational Problem In Potential Theory - Piano Class | MUS 31B and more Papers Piano in PDF only on Docsity! ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO 2000 Mathematics Subject Classification. 31B15, 31B20, 30C40, 47A75. Key words and phrases. Newtonian potential, Neumann-Poincaré operator, sym- metrizable operator, Poincaré’s variational problem, Schiffer’s operator, Hilbert- Beurling transform, Fredholm spectrum. 1 2 Abstract. One of the earliest attempts to rigorously prove the solvability of Dirichlet’s boundary value problem was based on seeking the solution in the form of a ”potential of double layer”, and this leads to an integral equation whose kernel is (in general) both singular and non-symmetric. C. Neumann succeeded with this ap- proach for smoothly bounded convex domains, and H. Poincaré, by a tremendous tour de force, showed how to push through the analysis for domains with sufficiently smooth boundaries but no hypothesis of convexity. In this work he was (according to his own account) guided by consideration of a variational problem involving the partition of energy of an electrostatic field induced by charges placed on the boundary of a domain, more precisely the charge dis- tributions which render stationary the energy of the field inside the domain divided by the energy of the field outside the domain. Un- fortunately, a rigorous treatment of this problem was not possible with the tools available at that time (as Poincaré was well aware). So far as we know, the only one to propose a rigorous treatment of Poincarés problem was T. Carleman (in the two dimensional case) in his doctoral dissertation. Thanks to later developments (especially concerning Sobolev spaces, and spectral theory of oper- ators on Hilbert space) one can now give a complete, general and rigorous account of Poincaré’s variational problem, and that is the object of the present paper. ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 5 for integral operators were not applicable. Nonetheless, the few ex- amples that were understood indicated that this integral operator had lots of real eigenvalues, as well as (non-orthogonal) eigenfunctions, and there was no general theory available that could explain such behav- ior. Later, Marty and Korn introduced the notion of symmetrizability of an operator, and showed that this applied to the N - P operator. But, over a decade before this happened Poincaré initiated the study of a certain (self adjoint!) variational problem that did not seem prima facie to be related to the N - P kernel, but turns out to be the ”high ground” which fully clarifies the ”self adjoint features” of the N - P integral operator. So far as we know this fascinating approach has not yet been fully worked out. To do so is our purpose in this paper. C. Neumann had proven the solvability of Dirichlet’s problem in convex domains by a recursive (and in principle constructive) procedure based on calculating an infinite sequence of double layer potentials which were the summands in a series converging to the solution. Later Poincaré had proven by an altogether different method (which he called ”balayage”) the solvability in domains of quite general character. Thus, in 1897, when Poincaré’s paper [34] appeared, there was already a rigorous proof at hand, but Poincaré set himself the task to find an alternate proof based on establishing the convergence of the Neumann 6 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO series also for nonconvex domains. This he succeeded to do (under fairly strong regularity assumptions: the domain had a C2 boundary and the boundary values were sufficiently differentiable). Motivation for this undertaking was that the balayage method was not suitable for numerical computation of solutions, and a solution based on Neumann’s series was superior in this regard. Poincaré’s new proof was extremely long and technical, and we won’t enter into the details here. It is based on energy estimates for the elec- trostatic field due to charges distributed on the boundary of a domain, more precisely how the energy is partitioned between the part of the field inside the domain and the part lying outside. The final section of the paper has an unusual character: Poincaré poses an extremal problem (more precisely, a sequence of such problems) concerning this partition of energy, which he says guided his steps through the pre- ceding demonstration. But this is placed in a sort of quarantine , and not referred to in the course of the demonstration because, as Poincaré repeatedly tells us, he has been unable to establish the salient details of the extremal problem on a rigorous basis. The paper concludes with the words: ”After having established these results [concerning conver- gence of the Neumann series] rigorously, I felt obliged in the two final chapters to give an idea of the insights which initially led me to foresee ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 7 these results. I thought that, despite their lack of rigor, these could be useful as tools for research insofar as I had already used them success- fully once.” Here is what is involved. Consider (we use here, for the most part, Poincaré’s notations) a closed surface Γ in R3 and denote by Ω,Ω′ respectively the interior and exterior domains into which R3 is parti- tioned by Γ. Let there be given a real valued continuous function u on R3 and whose restrictions to Ω and Ω′ (denoted by W and W ′) are harmonic functions with finite Dirichlet integrals (denoted J(W ) and J(W ′) ). Then u is the potential of an electrostatic charge distributed on Γ. (In modern language, this charge is f := ∆u in the distributional sense, it is a Schwartz distribution with support in Γ.) If we assume the total energy J(W ) + J(W ′) equals 1, what is the minimum value possible for J(W )? It is 0, and this is attained if, and only if W is a constant c, and W ′ is the solution to the Dirichlet problem for the exterior domain with data W ′ = c on Γ (the ”conductor potential” of Γ). Here c is to be chosen so that J(W ′) = 1. The correspond- ing charge distribution f is (modulo a constant factor) the equilibrium measure for the compact set Γ. All this was well understood at the time. But now Poincaré embarks into terra incognita: Consider the 10 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO Coming back to Poincaré’s problem seen in this light, the immedi- ate question is: Is there a compact linear operator lurking behind the form J? Yes, there is, but we must first replace J by J − J ′ (which clearly leads to an extremal problem equivalent to the former insofar as the ratios J/(J + J ′) and (J − J ′)/(J + J ′) ...or for that matter J/J ′ , which Poincaré actually uses...are simply related. It is a highly nontrivial fact that, restricted to the subspace P of H consisting of pairs (V, V ′) with equal traces on Γ, the form J −J ′ is completely con- tinuous (to use an older terminology, that held sway when ”operators on Hilbert space” were exhibited in terms of Hermitian forms rather than operators). This was first established rigorously by T. Carleman in his remarkable doctoral dissertation [4]. Following modern practice we shall, below, rework all this in the language of operators along the lines of the abstract treatment given by M. G. Krein [20]. This gets us off the ground: extrema in Poincaré’s problem are al- ways attained. In terms of the abstract model, we can continue to seek (and find) the minima of 〈Tx, x〉 subject to the successively stricter orthogonality constraints imposed on the unit vector x. But one point has to be emphasized: If we assume (as is the case in Poincaré’s prob- lem) that 〈Tx, x〉 takes negative values it attains a minimum on the unit sphere. In general, for any compact operator T, 〈Tx, x〉 attains a ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 11 maximum and a minimum on the unit ball but only in the case of a positive maximum or a negative minimum can we assert the extremal element has norm 1. Thus, the sequence of minimum problems will continue to furnish an increasing sequence of negative eigenvalues of T so long as 〈Tx, x〉 attains negative values for x among the remaining (competing) vectors. If this is not so, the process terminates, and ei- ther all remaining x are in the kernel of T , or 〈Tx, x〉 takes positive values for some x. If this is the case, we can maximize 〈Tx, x〉 among all unit vectors, and then, analogously as before continue to find a de- creasing sequence of positive eigenvalues (and associated eigenvectors) of T , which process only terminates if at some point 〈Tx, x〉 takes no positive values on the eligible set of x. From modern spectral theory we know moreover that to each negative and each positive eigenvalue is associated only a finite dimensional family of eigenvectors, whereas cor- responding to the spectral point 0 there may be either no eigenvector, or a family of finite or infinite dimension. Poincaré seems to have conjectured that (translated into our ter- minology) an infinite sequence of increasing negative eigenvalues (the first being −1) would exist (i.e. his recursive process would never ter- minate ) and that moreover the associated eigenfunctions would span the Hilbert space. This is a very bold conjecture, implying that the 12 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO operator associated to the form J ′ − J(which we shall later relate to the so-called Neumann - Poincaré integral operator) has only positive spectrum and moreover is injective. These assertions are true in case Γ is a sphere, but we will show they do not hold generally, indeed not even for ellipsoids of revolution in R3. For d = 2 there are some notable anomalies. To complete this survey of Poincaré’s extremal problem we should take up his variational condition for extrema, which characterizes the extremal potentials u = (W,W ′) by the condition that the normal derivatives of W and W ′ , computed along Γ from opposite sides with respect to the same normal vector, are a (negative) constant multiple of one another. We postpone the further examination of this con- dition, which relates to the aforementioned integral operator and its symmetrization, to a later section. Let us briefly describe the contents of the paper. Section 2 contains some terminology and conventions plus a collection of known facts from the Newtonian potential theory, seen from the modern point of view of distribution theory and Sobolev spaces. Section 3 is devoted to an ab- stract symmetrization principle for linear bounded operators acting on a Hilbert space. This theme was popular precisely because of potential ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 15 Acknowledgements. The authors are indebted to the Mathemati- cal Research Intitute at Oberwolfach, Germany (the Research in Pairs Programme) for support and excellent working conditions. The first named authors gratefully acknowledge the support from the National Science Foundation. 2. Prerequisites of potential theory The aim of this section is to assemble some terminology and basic facts of Newtonian potential theory. Let Ω be a bounded domain in Rd with boundary Γ. We assume that Γ is at least C2-smooth. The (d− 1)-dimensional surface measure on Γ is denoted by dσ and the unit outer normal to a point y ∈ Γ will be denoted ny. Throughout this article E(x, y) = E(x − y) denotes the normalized Newtonian kernel: E(x, y) =  1 2π log 1|x−y| , d = 2, cd|x− y|2−d, d ≥ 3, (1) where c−1d is the surface area of the unit sphere in R d. The signs were chosen so that ∆E = −δ (Dirac’s delta-function). For a C2-smooth function (density) f(x) on Γ we form the funda- mental potentials: the single and double layer potentials in Rd; denoted 16 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO Sf and Df respectively: Sf (x) = ∫ Γ E(x, y)f(y)dσ(y) Df (x) = ∫ Γ ∂ ∂ny E(x, y)f(y)dσ(y). (2) The surface Γ divides Rd into two domains Ω = Ωi (interior to Γ) and the exterior Ωe. Thus the potentials above define pairs of functions (Sif , S e f ) and (D i f , D e f ) which are harmonic in Ωi and Ωe respectively. As is well known from classical potential theory (cf. [15, 42]) denot- ing by Sif (x), ∂ ∂nx Sif (x) (and corresponding symbols with superscript e) the limits at x ∈ Γ from the interior (or exterior), the following relations (known as the jump formulas for the potentials) hold for all x ∈ Γ: Sif (x) = S e f (x); ∂ ∂nx Sif (x) = 1 2 f(x) + ∫ Γ ∂ ∂nx E(x, y)f(y)dσ(y); ∂ ∂nx Dif (x) = ∂ ∂nx Def (x); Dif (x) = − 1 2 f(x) + ∫ Γ ∂ ∂ny E(x, y)f(y)dσ(y); ∂ ∂nx Sef (x) = − 1 2 f(x) + ∫ Γ ∂ ∂nx E(x, y)f(y)dσ(y); Def (x) = 1 2 f(x) + ∫ Γ ∂ ∂ny E(x, y)f(y)dσ(y). (3) ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 17 We warn the reader that different conventions (on the choice of the sign of the fundamental solution or the unit normal) may affect these formulas. Rather direct computations ( see for instance Chapter II in [15], Chapters 18-19 in [42] or [29]) show that the integral kernels K(x, y) := − ∂ ∂ny E(x− y); K∗(x, y) = − ∂ ∂nx E(x− y) satisfy on Γ the following estimates, for d ≥ 3: |K(x, y)| = O( 1 |x− y|d−2 ), x, y ∈ Γ, x 6= y, |K∗(x, y)| = O( 1 |x− y|d−2 ), x, y ∈ Γ, x 6= y. (4) For d = 2, due to the fact that log |z−w| is the real part of a complex analytic function log(z−w) = log |z−w|+ i arg(z−w), z, w ∈ Γ, and by Cauchy-Riemann’s equations one obtains K(z, w) = ∂ ∂τw arg(z − w), where τw is the unit tangent vector to the curve Γ. Thus, on any smooth curve Γ ⊂ R2, the kernels K(z, w) and K∗(z, w) are uniformly bounded. 20 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO Only locally constant functions are annihilated by this seminorm. It will be necessary to distinguish between the two restrictions of h to the inner and outer domain; we denote h = (hi, he) where hi = h|Ωi and similarly he = h|Ωe . In virtue of Poincaré’s inequality the functions hi and he are in the Sobolev W 1,2-spaces of the corresponding domains. To simplify notation we put henceforth W s = W s,2. We can regard an element h ∈ H ⊂ D′(Rd) as a distribution defined on the whole space. Then ∆h = µ ∈ D′Γ(Rd) (the lower index means supp(µ) ⊂ Γ) and, by a slight abuse of notation −h(x) = Sµ(x) = ∫ Γ E(x, y)dµ(y), x ∈ Rd \ Γ. (8) If the distribution µ is given by a smooth function times the surface measure of Γ, then h = Sµ and by (6) hi|Γ = he|Γ. Our next aim is to identify the closed subspace of H characterized by the latter matching property. By assumption the surface Γ is smooth. Hence there are linear con- tinuous trace operators Tr : W 1(Ωi,e) −→ W 1/2(Γ). Moreover, the trace operator from each side of Γ is surjective (and hence it has a continuous right inverse), see for instance [27]. We will ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 21 denote in short h|Γ = Tr h. If d ≥ 3, then for any function f ∈ W 1/2(Γ) there exist solutions (hi, he) ∈ H to the inner and outer Dirichlet problems with boundary data f : hi|Γ = he|Γ = f , see [24]. In the case d = 2 the additional assumption ∫ Γ fdσ = 0 must be made, to assure the existence of he with he(∞) = 0 and finite energy, see [24] . The following consequence of Green’s formula will be frequently used in this section. For a harmonic function u in Ω, of class C2 on the closed domain: 2 ∫ Γ u ∂u ∂n dσ = ∫ Ω ∆(u2)dx = 2 ∫ Ω |∇u|2dx. (9) Another common form of Green’s formula, for arbitrary functions φ, ψ ∈ C2(Ω) reads: ∫ Ω ∇φ · ∇ψdx+ ∫ Ω φ∆ψdx = ∫ Γ φ∂nψdσ. As a first application we note an important isometric identification, see [24]. Lemma 2.1. Let f ∈ L2(Γ). Then 〈Sf, f〉 = ‖Sf‖2H. (10) 22 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO We deduce from here that S is a non-negative self-adjoint operator on L2(Γ). Moreover Sf = 0 implies ∇Sf = 0 in Rd \ Γ, whence Sf is constant on both sides of Γ. But Sf (∞) = 0, so Sf = 0 as a distribution on Rd. Therefore f = −∆Sf = 0. This proves that S is a strictly positive operator on L2(Γ). We will prove below that S is not invertible. Proposition 2.2. Assume d ≥ 3 and let h = (hi, he) ∈ H. Then hi|Γ = he|Γ if and only if there exists ρ ∈ W−1/2(Γ) such that h = Sρ. Proof. Let H− be the completion of L 2(Γ) with respect to the Hermit- ian form 〈Sf, f〉 = ‖ √ Sf‖2. Let H+ = Ran √ S, viewed as a non-closed vector subspace of L2(Γ), and also regarded as the domain of the posi- tive unbounded operator √ S −1 . Note that H+ is a complete space with respect to the norm induced by the form 〈 √ S −1 f, √ S −1 f〉. Then the positive operator S can be extended by continuity to an isomorphism S : H− −→ H+, and the L2-pairing 〈 √ Sf, g〉 = 〈f, √ Sg〉 defines a duality between the Hilbert spaces H+ and H−. The above standard duality construction can be correlated to the Dirichlet space seminorm of H. First we polarize the identity in the ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 25 formula yields | ∫ Γ ∂hi ∂n gidσ| = | ∫ Ω ∇hi · ∇gidx| ≤ ‖∇hi‖2,Ω‖∇gi‖2,Ω. By Banach’s open mapping theorem, the continuous bijective trace operator Tr : {h ∈ W 1(Ω),∆h = 0} −→ W 1/2(Γ) is bicontinuous, hence, in our situation we find a positive constant C such that ‖∇gi‖2,Ω ≤ C‖gi|Γ‖W 1/2 . Consequently | ∫ Γ ∂hi ∂n gidσ| ≤ C‖∇hi‖2,Ω‖gi‖W 1/2 . A standard regularization argument shows that every harmonic func- tion gi in Ω, having finite energy inside Ω (i.e. ‖∇gi‖2,Ω < ∞) can be approximated in the energy metric by harmonic functions which are smooth up to the boundary. And we know that the traces gi exhaust the space W 1/2(Γ). Thus ∂hi ∂n dσ defines a linear continuous functional on W 1/2(Γ), which via the L2(Γ) duality can be identified with a dis- tribution ∂hi ∂n ∈ W−1/2(Γ). Moreover, the above estimate implies ‖∂hi ∂n ‖W−1/2(Γ) ≤ C‖∇hi‖2,Ω. Again by regularization we obtain the following statement. 26 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO Proposition 2.3. The normal derivatives of the boundary values of a pair (hi, he) ∈ H are distributions ∂hi∂n , ∂he ∂n ∈ W−1/2(Γ) depending continuously on ‖hi‖H, ‖he‖H, respectively, and satisfying the duality identities: ∫ Γ ∂hi ∂n gidσ = ∫ Γ hi ∂gi ∂n dσ = ∫ Ω ∇hi · ∇gidx,∫ Γ ∂he ∂n gedσ = ∫ Γ ge ∂he ∂n dσ = − ∫ Ω ∇he · ∇gedx, (11) for every g = (gi, ge) ∈ H. We are ready to identify double layer potentials in the space H. Corollary 2.4. Let h = (hi, he) ∈ H. The following conditions are equivalent: a) h ∈ D(= S⊥); b) ∂hi ∂n = ∂he ∂n (in W−1/2(Γ)); c) There exists f ∈ W 1/2(Γ) such that h = Df . In this case f = he − hi. Proof. Assume that h, g ∈ H are orthogonal elements. Then the above proposition yields ∫ Γ ( ∂hi ∂n gi − ∂he ∂n ge)dσ = 0. ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 27 Any element f ∈ W 1/2(Γ) can be realized as f = gi = ge for a proper choice of g, hence b) follows. Conversely, if b) holds, then the same identity implies a). Assume that b) is true and define f = he−hi. ThenDf is well defined and satisfies by (6) (Df )e−(Df)i = f, ∂(Df )i∂n = ∂(Df )e ∂n . This proves that the pair of harmonic functions h−Df form a single harmonic function on Rd which vanishes at infinity, and hence identically. Therefore h = Df .  The only elements of H annihilated by the energy seminorm are scalar multiples of (1, 0). This is the double layer potential of the constant function, and is, therefore, orthogonal to S. By the boundary formula (Df )e = 1 2 f − 1 2 K we infer K1 = 1. Another distinguished element of H is provided by the equilibrium distribution ρ on Γ; namely (1, h) ∈ S, that is Sρ = 1 and h = Seρ, see Example 1. The following result is known as Plemelj’s symmetrization principle, see [33, 16, 18] . Lemma 2.5. The operators S,K : L2(Γ) −→ L2(Γ) satisfy the identity KS = SK∗. (12) 30 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO 〈f, g〉R = 〈Rf,Rg〉, f, g ∈ H. Note that the space H is complete with respect to this new norm if and only if the operator R is invertible. The assumption in the statement implies 〈Mf, g〉R = 〈RMf,Rg〉 = 〈Rf,RMg〉 = 〈f,Mg〉R. That is M is a symmetric operator with respect to the new scalar product. We prove a little more than the statement. Namely that there exists a bounded self-adjoint operator A ∈ Cp with the property: AR = RM. (13) Let N denote the self-adjoint operator: N = R2M = M∗R2. We regularize the inverse of R by a small positive parameter ; to the effect that the strong operator topology limit (R + )−1R → I exists when  tends to zero. And for any operator L ∈ Cq the limit (R + )−1RL → L exists in the norm topology of Cq (by a finite rank approximation argument). ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 31 Fix a positive integer n so that p < 2n and consider the operator: A = (R + ) −1R2M(R + )−1. Then A ∈ Cp ⊂ C2n, and |A|2 = A2 = (R + )−1R2M(R + )−2R2M(R + )−1. In virtue of the cyclic invariance of the trace we obtain: tr|A|2n = tr[(R + )−2R2M ]2n → trM2n <∞. Thus the family of operators (A)>0 is bounded in C2n, hence rela- tively compact in the weak topology of the same ideal. On the other hand, AR converges in the norm topology of C2n to RM . This implies that any weak limit point A of (A)>0 must satisfy the identity AR = RM . Since the operator R was assumed to be injective, all limit points coincide with a uniquely determined operator A ∈ C2n satisfying identity (1). Moreover, RAR = R2M = M∗R2 = N. (14) Since R is injective with dense range, it follows from 〈ARf,Rg〉 = 〈Rf,ARg〉, f, g ∈ H, that A is self-adjoint. 32 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO It remains to prove that A ∈ Cp. Let λk be a non-zero value in the spectrum of A and let fk be a corresponding eigenvector, normalized by the condition ‖fk‖ = 1. Then, by (13), M∗Rfk = RAfk = Rλkfk. Hence λk ∈ σ(M∗). Since M∗ ∈ Cp, the convergence of the series ∞∑ k=1 |λk|p <∞ shows that A ∈ Cp. In the enumeration λk we allow multiplicities and can assume by the compactness of A that |λk| ≥ |λk+1|, for all values of k. As for the zero eigenvectors h of A, they span its kernel, and each element Rh is annihilated by M∗, as follows from the identity M∗Rh = RAh = 0. Since the vectors (fk)k≥1 span, together with h ∈ kerA, the space H, the M∗-eigenvectors (Rfk)k≥1 and (Rh)Ah=0 span H by the density of the range of R. This proves the last part of the theorem. Remark next that the spectral decomposition of A provides a norm convergent series: Af = ∑ k λk〈f, fk〉fk, f ∈ H. (15) ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 35 Along the same lines, note that the self-adjoint operator N = RAR admits a non-orthogonal, norm convergent decomposition into rank- one self-adjoint operators (cf. (13)): Nh = ∑ k λk〈h,Rfk〉Rfk, h ∈ H. (16) It does not follow from (16) that the eigevectors of M are complete in H. 2. Compactness and eigenvalues in the negative space. Let H̃ be the completion of H in the ‖.‖R-norm, and let M̃ be the linear continuous extension of M there. We call H̃ the negative space by analogy with distribution theory (and the theory of Gelfand triples). Our next aim is to prove that M̃ is compact and self-adjoint on H̃ and that every eigenvector of M̃ corresponding to a non-zero eigenvalue belongs to H, and hence it is an eigenvector of M . The intertwining identity AR = RM implies that M̃ is compact on H̃ ( even in the same class Cp). We prove the compactness of M̃ . Let (φn) be a bounded sequence in H̃. We can find a sequence (hn) in H such that ‖R(φn − hn)‖ ≤ 2−n, n ≥ 1. Then by the compactness of A, the sequence (ARhn) has a convergent subsequence which we will denote by the same symbols (ARhn). But ARhn = RMhn. That is 36 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO the sequence Mhn is convergent in the ‖.‖R-norm. Thus the sequence (M̃hn) is convergent in H̃, and so is the sequence (M̃φn). We extend next the operator R to H̃. Let h̃ ∈ H̃ and consider a sequence (hn) in H which converges in H̃ to h̃. By the very definition of the negative norm, the sequence (Rhn) is Cauchy in H. Define R̃h = limnRhn. Moreover, this definition implies that the operator R̃ : H̃ −→ H is continuous in the respective norms, and ‖R̃h̃‖ ≤ ‖h̃‖R. Since the range of R is dense in H, it follows that R̃H̃ = H. On the other hand, we can consider R−1 : RH −→ H as an un- bounded self-adjoint operator. Its domain, H+ = RH is complete with respect to the graph norm ‖h‖H+ = ‖R−1h‖. Let L be a linear con- tinuous functional on H+. By the Riesz representation lemma, and by the preceding definition of R̃ there exists an element g = R̃g̃ ∈ H such that L(Rh) = 〈h, g〉 = 〈h, R̃g̃〉 = 〈Rh, h̃〉. Therefore, the scalar product of H defines a non-degenerate contin- uous pairing between the Hilbert spaces H+ = RH and H̃. We claim that M̃ is the adjoint of M with respect to this duality pairing. Indeed, let h̃ ∈ H̃ and f ∈ H+ be arbitrary elements. The above definitions ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 37 imply: 〈M̃h̃, f〉 = 〈R̃M̃h̃, R−1f〉 = 〈AR̃h̃, R−1f〉 = 〈R̃h̃, AR−1f〉 = 〈h̃, RAR−1f〉 = 〈h̃,M∗f〉. Consequently, by general duality theory for the spectral spaces, the spectrum of M̃ is real, equal to the spectrum of the compact operator M∗ : H+ −→ H+, and the multiplicities of the non-zero eigenvalues are equal. But every eigenvalue ofM∗ : H+ −→ H+ is an eigenvalue ofM∗ : H −→ H. And conversely, we have proved that every eigenvalue of M∗ is of the form R2gk, where Mgk = λkgk. Hence the spectral subspace corresponding to a non-zero eigenvalue of M∗ : H −→ H is included in H+, and has the same multiplicity as the spectral subspace of M . But the operator M is a restriction of M̃ to H and as such coincides with the latter on all finite dimensional subspaces of H. Therefore, all eigenvectors of M̃ corresponding to a non-zero eigenvalue belong to H, and coincide with the eigenvectors of M . For more details about the above proof, and another way of reaching the same conclusion, the reader may consult [20]. 3. Min-max. Having understood the full spectral picture of the operator M and its continuous extension M̃ we are now prepared to 40 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO Then, λ+k = max f⊥{g+0 ,...,g + k−1} 〈SMf, f〉 〈Sf, f〉 . (17) and similarly λ−k = min f⊥{g−1 ,...,g − k−1} 〈SMf, f〉 〈Sf, f〉 . (18) 4. Operators with a continuous kernel. A slightly stronger assump- tion than the S-symmetry of the compact operator M discussed above is the factorization M = LS, where S > 0 and L ≥ 0 is a compact operator. Indeed, SM = SLS = M∗S. This is the class of symmetrizable operators with a continuous kernel, in the terminology of Krein [20]. We put as before R = √ S. Since M = (LR)R, we find ‖Mf‖ ≤ ‖LR‖‖f‖R, f ∈ H. Therefore the continuous extension M̃ maps continuously and com- pactly the negative space H̃ into H. Henceforth we assume that S > 0 and L ≥ 0. As a consequence of the compactness and positivity of L one obtains the convergence in H of the series: Lf = ∑ k λk〈f, gk〉gk, f ∈ H, (19) ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 41 see Theorem 9 of [20]. Indeed, recall that (gk)k is an orthonormalized system of eigenvectors in H̃, which spans together with ker M̃ the whole space. In particular, for a vector f ∈ H we have the convergent Fourier series in H̃: f = ∑ k 〈f, Sgk〉gk + ∑ j 〈f, ξj〉H̃ξj, where (ξj) is a completion of (gk), with vectors in ker M̃ , to an or- thonormal basis. By applying the operator M to the above sum we find LSf = Mf = ∑ k λk〈Sf, gk〉gk, where the convergence is now assured in H. Let LNf = ∑ k≤N λk〈Sf, gk〉gk, so that 〈LSf, Sf〉 = ∑ k λk|〈Sf, gk〉|2 ≥∑ k≤N λk|〈Sf, gk〉|2 = 〈LNSf, Sf〉, f ∈ H. But the range of S is dense in H, so that LN ≤ L, as self-adjoint operators. Then it is well known that L′ = SOT − limN LN exists, it is a bounded operator and moreover L(Sf) = L′(Sf) for all f . In conclusion L = L′ and the convergence of the expansion (19) is proved. Assume in addition that the operator L is strictly positive. Then the system (gk) ⊂ H of eigenfunctions of M spans H, and at the same 42 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO time it is an orthonormal basis in H̃. Indeed, if 〈x, gk〉 = 0 for all k, then Lx = 0 and x = 0. The expansion (19) can be regarded as an abstract analogue of Mer- cer’s theorem in the theory of integral operators, see [20]. 5. The norm of a symmetrizable operator. Let L(H) denote the C∗-algebra of linear bounded operators acting on the Hilbert space H. Assuming R2M = M∗R2 as in Theorem 3.1 we immediately obtain the formula ‖M‖L(H̃) = λ + 0 . (20) Indeed,using the notation introduced in the proof of Theorem 3.1, 〈Mf, f〉H̃ = 〈ARf,Rf〉H ≤ λ + 0 ‖f‖2H̃ = λ + 0 ‖Rf‖2H , and the inequality is attained by the compactness of the operator A. For a symmetrizable operator M as before, the following non-trivial norm estimate holds: ‖M‖L(H̃) ≤ ‖M‖L(H), see [20]. ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 45 find ∫ Ωe |∇Sg|2dx = − ∫ Γ Sg ∂nS e gdσ = 〈Sg, 1 2 g + 1 2 K∗g〉2,Γ, and ∫ Ωi |∇Sg|2dx = ∫ Γ Sg ∂nS i gdσ = 〈Sg, 1 2 g − 1 2 K∗g〉2,Γ. Therefore, 〈(Pe − Pi)Sg, Sg〉H = 〈KSg, g〉2,Γ, ‖Sg‖2H = 〈Sg, g〉2,Γ.  In view of Plemelj’s symmetrization principle (Lemma 2.5), the con- ditions of the abstract symmetrization scheme in Theorem 3.1 are met for the second Rayleigh quotient above. Accordingly we can state the following theorem, whose main points were foreseen by Poincaré. Theorem 4.2. Let Ω ⊂ Rd be a bounded domain with smooth boundary Γ and let Ωe = R d \ Ω. Let Sρ denote the single layer potential of a distribution ρ ∈ W−1/2(Γ), (ρ(1) = 0 in case d = 2). Define successively, as long as the maximum is positive, the energy quotients λ+k = max ρ⊥{ρ+0 ,...,ρ + k−1} ‖∇Sρ‖22,Ωe − ‖∇Sρ‖ 2 2,Ω ‖∇Sρ‖22,Rd , (22) 46 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO where the orthogonality is understood with respect to the total energy norm. The maximum is attained at a smooth distribution ρ+k ∈ W 1/2(Γ). Similarly, define λ−k = min ρ⊥{ρ−1 ,...,ρ − k−1} ‖∇Sρ‖22,Ωe − ‖∇Sρ‖ 2 2,Ω ‖∇Sρ‖22,Rd . (23) The minimum is attained at a smooth distribution ρ−k ∈ W 1/2(Γ). The potentials Sρ±k together with all Sχ ∈ kerK(χ ∈ W−1/2(Γ)), are mutually orthogonal and complete in the space of all single layer potentials of finite energy. The stronger than expected regularity of the eigenfunctions (ρ±k ∈ W 1/2(Γ)) was explained in abstract form, in the last section. The equi- librium distribution of Ω provides the first function ρ+0 in this process: Sρ+0 = 1, Sρ+0 |Ω = 1. The first eigenvalue is always λ + 0 = 1 and has multiplicity equal to one (cf. Example 8.1). Lemma 4.1 gives a precise correlation between the above Poincaré variational problem and the Neumann-Poincaré operator. Corollary 4.3. The spectrum of the Neumann-Poincaré operator K, multiplicities included, coincides with the spectrum (λ±k ) of Poincaré’s variational problem, together with possibly the point zero. The extremal distributions for the Poincaré problem are exactly the eigenfunctions of K. ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 47 In practice it is hard to work directly with the N-P operator on L2(Γ). Instead, the following interpretation of the extremal solutions to Poincaré’s variational problem is simpler and more flexible. This also goes back to Poincaré’s memoir [34] , and it was constantly present in the works of potential theory in the first decades of the twentieth century, cf. for instance [33]. Let us start with an eigenfunction f ∈ L2(Γ) of the operator K∗. Then K∗f = λf ⇒ KSf = SK∗f = λSf, and by the jump formulas (6) ∂nS i f = 1− λ 2 f, ∂nS e f = −1− λ 2 f. The associated energies are Ji[f ] = ∫ Ωi |∇Sf |2dx = 1− λ 2 〈Sf, f〉, Je[f ] = ∫ Ωe |∇Sf |2dx = 1 + λ 2 〈Sf, f〉. To verify our computations, simply note that Je[f ]− Ji[f ] Je[f ] + Ji[f ] = λ. The characteristic feature of the above single layer potential Sf is encoded in the following statement. 50 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO jump formulas (6) and (25) we find 〈(Pd − Ps)(Df + Sg), Df + Sg〉H = ∫ Ω ∇(Df − Sg) · ∇(Df + Sg)dx = ∫ Γ (Dif−Sig)∂n(Dif+Sig)dσ = ∫ Γ (−1 2 f−1 2 Kf−Sg)∂n(Dif+Sig−Def−Seg)dσ = − ∫ Γ Kfgdσ. Similarly, ‖Df + Sg‖2H = ∫ Ω |∇(Df + Sg)|2dx = − ∫ Γ fgdσ. We have arrived at the following isometric identification. Lemma 5.1. Let h ∈ H be an element supported by the inner domain, i.e. he = 0. Decompose h = Df + Sg, f ∈ W 1/2(Γ), g ∈ W−1/2(Γ). Then g, f ⊥ 1, (I −K)−1Sg is well defined, and 〈(Pd − Ps)h, h〉H ‖h‖2H = 〈K(I −K)−1Sg, g〉2,Γ 〈(I −K)−1Sg, g〉2,Γ . (26) To put this into the abstract symmetrization scheme we have only to replace L2(Γ) by the codimension one subspace H = 1⊥ of vectors orthogonal to the constants. The operator (I −K)−1S is strictly posi- tive on H, and can replace S in Proposition 3.2. However, in general, the operator K does not leave H invariant. To correct this we con- sider the orthogonal projection PH of K onto H and the compression ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 51 K1 = PHKPH of K; then K1(I −K)−1S = (I −K)−1SK∗1 . Indeed, start with f, g ∈ H satisfying f = (I −K)−1Sg, that is (I − K)f = Sg. Then (I − K)Kf = KSg = SK∗g, or equivalently (I − K)K1f = SK ∗ 1g, which is the relation to be proved. The following analogue of Poincaré’s principle holds. Theorem 5.2. Let Ω ⊂ Rd be a bounded domain with smooth bound- ary. Let Ps, Pd denote the orthogonal projections of the energy space H onto the subspace of single, respectively double layer potentials. Let Hi be the subspace of functions vanishing on the complement of Ω. Define successively, as long as the maximum is positive, the energy quotients λ+k = max h∈Hi h⊥{h+1 ,...,h + k−1} 〈(Pd − Ps)h, h〉Hi ‖h‖2Hi . (27) Then the maximum is attained at an element h+k ∈ Hi. Similarly, define λ−k = min h∈Hi h⊥{h−1 ,...,h − k−1} 〈(Pd − Ps)h, h〉Hi ‖h‖2Hi . (28) The minimum is attained at h−k ∈ Hi. Exactly as before, the link to the operator K is very simple. 52 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO Corollary 5.3. In the conditions of the Theorem, the spectrum of the Neumann-Poincaré operator K consists, including mutiplicities, of {λ±k ; k ≥ 1} together with the points {0, 1}. Remark that the eigenvalue λ+0 = 1 cannot be detected by the above variational scheme. This is due to the fact that the corresponding eigenfunction 1 of K cannot satisfy the compatibility condition −1 2 1 + 1 2 K1−Sg = 0, (which would mean g = 0). This scenario would produce the pair (1, 0) of zero total energy. The extremal solutions to the above problem are precisely h±k = Su±k +Dg±k , where Kg±k = λ ± k g ± k , and 2Su±k = (1− λ ± k )g ± k . Next we describe a realization of the abstract angle operator Pi(Pd− Ps)Pi. To this aim we consider an arbitrary element h ∈ H and its harmonic field ∇h ∈ L2(Rd, dx). We define, for points x ∈ Ω Π(∇h)(x) = p.v.∇x ∫ Rd ∇yE(x, y) · ∇yhdy. (29) ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 55 Proof. For the proof we have only to observe the validity of the identity 〈TΩ(∇u),∇u〉 ‖∇u‖2 = 〈K1(I −K)−1Sg, g〉2,Γ 〈(I −K)−1Sg, g〉2,Γ , where (u, 0) = Df + Sg. The point 1 is missing from the spectrum of TΩ because (u, 0) = Sg + Df imples f, g ∈ H, that is f, g ⊥ 1 and the compression of K to the space H eliminates the point 1 from the spectrum, cf. the text following (26).  Since the operator TΩ is invariant under homotheties x 7→ tx, t > 0, we deduce that the spectrum of the Neumann-Poincaré operator asso- ciated to a domain Ω is invariant under all shape preserving transfor- mations (i.e., translations, rotations and homotheties) of Ω. 6. Neumann-Poincaré’s operator in two dimensions The natural connection to complex analytic functions provides a bet- ter understanding of the spectral analysis of the Neumann-Poincaré operator in two dimensions. There are a few specific two dimensional phenomena, whose discovery goes back to the works of Ahlfors [1] , Bergman [2, 3], Plemelj [33], Schiffer [40, 41] and Springer [43]. This 56 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO section is devoted to the proofs of some of the specifically two dimen- sional results which are related to the main theme of the present article. We do not aim at completeness, and for example we do not discuss the link between the eigenvalues of the Neumann-Poincaré operator and quasiconformal mappings. These and other results are well exposed in the aforementioned works. On the other side none of these papers em- phasizes on the relationship between the spectrum of the N-P operator and Poincaré’s extremum problem, the focus of the present study. We return to the notation introduced in the preliminaries, with some specific adaptations to dimension two: Γ is a C2-smooth Jordan curve, surrounding the domain Ω ⊂ C, and having Ωe as exterior domain. We denote by z, w, ζ, ... the complex coordinate in C and by ∂z = ∂ ∂z the Cauchy Riemann operator, and so on. The area measure will be denoted dA. The space H consists of (real-valued) harmonic functions h on C \ Γ having square summable gradients: h ∈ H ⇔ ∫ Ω∪Ωe | ∂zh(z)|2dA(z) <∞, h(∞) = 0. Note that the gradients ∂zh are now square summable complex anti- analytic functions. In other terms, in our notation B(Ω) is the complex conjugate of the Bergman space A2(Ω) of Ω. ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 57 The single and double layer potentials are in this case strongly related to Cauchy’s integral. For instance, (Kf)(z) = ∫ Γ f(ζ)Re [ dζ 2πi(ζ − z) ] = 1 2π ∫ Γ f(ζ) d arg(ζ − z), see [15]. The following complex antilinear singular integral operator plays the role of the symmetry Pd − Ps in our notation. Let F = ∇Sf , f ∈ W 1/2(Γ), be regarded as a single anti-analytic function defined on all Ω ∪ Ωe. Define the Hilbert (sometimes called Beurling) transform (TF )(z) = p.v. 1 π ∫ Ω∪Ωe F (ζ) (ζ − z)2 dA(ζ) (30) Lemma 6.1. Let h ∈ H be represented as h = Df+Sg, f ∈ W 1/2(Γ), g ∈ W−1/2(Γ). Then T∇(Df + Sg) = ∇(Df − Sg). (31) The proof is very similar to the proof of Lemma 5.4 and we omit it. In other terms, returning to our old notation: T∇h = ∇(Pd − Ps)h, h ∈ H. In particular we note the following simple but important fact. Corollary 6.2. The antilinear transform T is an isometric isomor- phism of the space B(Ω)⊕B(Ωe) onto itself. 60 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO Proof. Let λ ∈ σ(K) \ {1} and let (u, 0) ∈ H be the associated eigen- function of the operator Pi(Pd − Pe)Pi, cf. Proposition 5.5. By the above correspondence there exists an anti-analytic function F = ∂zu satisfying TΩF = λF . Let G = iF and remark that the antilinearity of TΩ implies TΩG = −λG. Remark also that G = ∂zũ, where ũ is the harmonic conjugate of u. Thus, the eigenvector in H corresponding to the eigenvalue −λ is simply (ũ, 0).  The eigenvalue 1 does not have a companion, see Example 8.1. Another symmetry is available from the above framework. Proposition 6.5. Let Ω be a bounded planar domain with C2-smooth boundary and let Ωe be the exterior domain. Then the Bergman space operators TΩ and TΩe have equal spectra. Proof. Let (F, 0) be an eigenvector of TΩ, corresponding to the eigen- value λ. Denote T (F, 0) = (λF,G). Since T 2 = I we get (F, 0) = λT (F, 0) + T (0, G) = (λ2F, λG) + T (0, G). Thus T (0, G) = ((1 − λ2)F,−λG). This means −λ ∈ σ(TΩe) and by the preceding symmetry principle λ ∈ σ(TΩe).  ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 61 7. Qualitative analysis of the Neumann-Poincaré operator The present section is devoted to a few aspects of spectral analysis of the Neumann-Poincaré operator, obtained via Poincaré’s variational principles. Our first aim is to prove the existence of a domain in R3 which carries a negative spectrum (of the associated N-P operator), arbitrarily close to −1. This infirms Poincaré’s guess, based on the case of a ball, that the spectrum of spatial bodies is always non-negative. We start by constructing a couple of cut-off functions. Lemma 7.1. For every positive δ there exists an odd C∞-function ψ : R −→ R, such that ψ(t) = t, t ∈ [−1, 1],∫ |t|>1 |ψ′(t)|2dt < δ, lim ±t→∞ ψ(t) = ±1. Proof. Let Λ be a positive constant. Choose, for t > 1 ψ(t) = 1 + (t− 1)e−Λ2(t−1)2 . Then ψ′(t) = e−Λ 2(t−1)2 [1− 2(t− 1)2Λ2], 62 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO and the conditions in the statement are met for Λ sufficiently large.  Lemma 7.2. Given r > 1 there exists a function φ : R −→ [0,∞), such that φ(x) =  1 |x| ≤ 1, 0 |x| ≥ r2 and ∫ ∞ −∞ φ′(x)2dx = C r2 − 1 , for a universal constant C. Proof. Let χ be a smooth function satisfying χ(x) = 1 for x < 0 and χ(x) = 0 for x ≥ 1. Define φ(x) =  1 |x| ≤ 1, χ(| x−1 r2−1 |), |x| ≥ 1. Then φ′(x) = 1 r2 − 1 χ′( x− 1 r2 − 1 ), x > 1, and similary for x < −1. A change of variable in the integral will prove the statement.  Theorem 7.3. There exists a bounded domain with smooth boundary such that its Neumann-Poincaré operator has negtive spectrum, arbi- trarily close to −1. ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 65 This proves that the distribution u on the boundary of Ω produces an arbitrarily small energy ratio J ′[u]/J [u]. According to Poincaré’s principle, the associated N-P operator has then a point in the spectrum arbitrarily close to the value −1.  The above proof has a plausible physical interpretation: If a con- denser consisting of two neighboring parallel plates is charged by plac- ing large charges of equal magnitude and opposite signs on the plates, most of the energy of the resulting field is in the space between the plates. The next result complements the previous example. Theorem 7.4. Let Ω be a domain with smooth boundary Γ in R3. Then, there exists a positive constant c such that, if λ is an eigenvalue of the Neumann-Poincaré operator associated to Ω with λ < −1 + c and f a corresponding eigenfunction, f takes both positive and negative values on ∂Ω. The constant c can be chosen uniformly for all domains Ω with C2 boundary having uniformly bounded principal curvatures. Proof. We use c1, c2, ... to denote positive numerical constants. The proof is by contradiction. Suppose then that f is a non-negative eigen- function associated to eigenvalue λ < −1+c. We’ll show that for small 66 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO c this leads to a contradiction. We may assume w.l.o.g. that ∫ Γ fdσ = 1. (33) We first show J [f ] ≤ c1. (34) Indeed, we have λf(x) = ∫ Γ K(x, y)f(y)dσ(y), where K denotes the Neumann - Poincaré kernel. By iteration we get λ3f(x) = ∫ Γ K3(x, y)f(y)dσ(y) (35) where K3 is the third iterate of K. It is known that |K3(x, y)| ≤ c2 on Γ× Γ (see [15] ). Hence, in view of (35), and since |λ| cannot be small due to our assumption, f(x) ≤ c3, x ∈ Γ. (36) This immediately implies (34) . To proceed with the proof, note that we may (and do) assume Ω is contained in the ball B1/2 (where BR denotes the ball of radius R centered at 0). Let F be the function on ∂B1 such that the measure Fdσ is the balayage of fdσ . Then, P (x, y) denoting Poisson’s kernel for the unit ball (where x is in B1 and y in ∂B1 ), F (y) = ∫ Γ P (x, y)f(x)dσ(x). ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 67 Since P (x, y) ≥ c4 for |x| ≤ 1/2 (and hence for x in ∂Ω) this implies F (y) ≥ c4, y ∈ ∂B1. (37) Let h denote the function identically equal to 1 on ∂B1. It is im- portant to stress that h is a multiple of the equilibrium potential and J(h) > 0. Since F ≥ c4h, we have J [F ] ≥ (c4)2I[h] ( the energy functional is monotonic for positive charges) , and so J [F ] ≥ c5. (38) Now, from (38) follows that, for the electrostatic field in R3 engen- dered by F , the part in the exterior of B1 has energy greater than (1/2)c5 (this is a consequence of the theory of Poincaré’s variational problem for the ball, as presented in Section 8.2). Since F arises from f via balayage, this is identical (outside B1) with the field due to f , and so, a fortiori , the energy Je[f ] of the field due to f outside Ω is not less than (1/2)c5 =: c6. Recall now from (34) that J [f ] ≤ c1. Since λ = Je[f ]− Ji[f ] Je[f ] + Ji[f ] = 2 Je[f ] J [f ] − 1 ≥ 2(c6/c1)− 1 = c7 − 1, whereas we assumed λ < −1 + c. This is a contradiction for c = c7, and the proof is concluded.  Remark. By a similar argument one can show e.g. if f, g are eigen- functions each associated to some eigenvalue in the range (−1,−1+ c), 70 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO On the other hand, the boundary ∂Ω1 carries a Lebesgue space, the N-P operator K1 and the same cut-off projector by the characteristic function of Γ1(η), still denoted P1(η). Moreover, the isometric identifi- cation ‖P1(η)f‖∂Ω = ‖P1(η)f‖∂Ω1 , holds. Note that the projectors P1(η) converge strongly to the identity of L2(∂Ω1) when η converges to zero. Since the operator K1 is compact, the following norm convergence lim η→0 ‖K1 − P1(η)K1P1(η)‖ = 0, is true. As a conclusion of these computations, and the general approxima- tion principle stated at the beginning of the proof, we find that for η sufficiently small the spectrum of P1(η)K1P1(η) approaches within distance  the given finite part F1 of the spectrum of K1. Similarly, the projection P2(η) does the same service on the boundary of Ω2 +Ra. Thus, the two ”corners” P1(η)KP1(η) and P2(η)KP2(η) of the N-P operator of Ω have spectra in an  neighborhood of the given set F1∪F2. Next we use Poincaré’s theorem, asserting that the spectrum of K can equally be computed via the energy form 〈SK∗f, f〉 = 〈KSf, f〉, ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 71 and the freedom to choose the translation parameter R large. Indeed, lim R→∞ 〈SP1(η)f, P2(η)〉 = 0 for every pair of functions f, g ∈ L2(∂Ω). Likewise, using the energy norm interpretation of the boundary scalar product 〈Sf, f〉 we infer lim η→0 〈S(P1(η) + P2(η))f, f〉 = ‖f‖2. Since the operator K is compact, we deduce that the differences KS − (P1(η)KSP1(η)⊕ P2(η)KSP2(η)) and [K − (P1(η)KP1(η)⊕ P2(η)KP2(η))]S tend to zero uniformly, as soon as R becomes large and η small. Thus, the spectrum of K can be approximated in the specified sense by the spectrum of P1(η)KP1(η)⊕P2(η)KP2(η) and this completes the proof of the theorem.  Knowing that there are domains with negative spectrum as close to −1 as we desire, and the example of the unit ball (see Section 8.2), the preceding theorem shows that there are domains in Rd with at least as many finite negative and finite positive eigenvalues as one desires. 72 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO 8. Examples 8.1. The equilibrium distribution. The notations are those adapted in the preliminaries. Let (1, h) ∈ H be a single layer potential of the equilibrium distribution ρ ∈ W−1/2(Γ). Then by taking boundary val- ues along Γ we find: Siρ = S e ρ = 1, 0 = 2∂nS i ρ = ρ−K∗ρ. Therefore K1 = KSρ = SK∗ρ = Sρ = 1. If another function f ∈ W 1/2(Γ) satisfies Kf = f , then there exists ξ ∈ W−1/2(Γ) such that Sξ = f and by reversing the above identities we find K∗ξ = ξ, that is Sξ produces zero energy inside Ω, hence it is a constant function. But this will imply that ξ is a scalar multiple of the equilibrium distribution. Thus, for any closed smooth surface Γ, dim ker(K − I) = 1. On the other hand, always ker(K + I) = 0. Indeed, assume that Kξ + ξ = 0. That means K∗Sξ + Sξ = 0, that is ∂nS e ξ = 0, which means that the field ∇Sξ has zero energy on Ωe. Thus Seξ = 0 which implies ξ = 0. 8.2. The ball in Rd. The complete solution of Poincaré’s variational problem for the unit ball in R3 was given by Poincaré [34] . As this ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 75 For the disk in R2 a degeneracy occurs: For each k > 0, the space of ”spherical harmonics” of order k is 2 - dimensional, spanned by sin(kt) and cos(kt) , where t is an angle variable along the unit circle, and for each f in the codimension one span of these, the field it engenders has equal energies inside and outside the unit circle. Here the Neumann - Poincaré operator has rank one. It is known [39] that the disk is the only planar domain for which the N - P operator has finite rank. It is not known whether there are such domains in higher dimensions. Another characteristic property of the ball is discussed below. Theorem 8.1. The following is true: for a ball in Rd the N - P kernel is symmetric, and balls are the only domains with this property. Proof. For the proof let us confine attention to (smoothly bounded) domains in R3. The argument is nearly identical in all dimensions. Apart from a constant of normalization the kernel in question is K(x, y) = (x− y) · n(y) ‖x− y‖3 , x, y ∈ Γ, where C denotes the boundary of the domain Ω under consideration, and n(y) denotes the unit outer normal to Γ at y. The symmetry of K means (x− y) · n(y) = (y − x) · n(x), for all x, y ∈ Γ, 76 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO i.e. (*)For any two distinct points x, y of Γ the vector sum of the unit outer normals to Γ at x and y is perpendicular to the chord joining x and y. It is easy to check that spheres enjoy this property, so let us turn to the converse. Assume Γ has property (*). We shall show it is a sphere. First note the following two immediate consequences of (*): (i) If the normal to Γ at some point has other intersections with Γ, it coincides with the normal at each of those points. (ii) If the normals to Γ at two distinct points x, y intersect at a point z, the distances of z from x and y are equal. (Indeed , it follows from (*) that the triangle formed by x, y, z is isosceles, having equal angles at x and y.) We now conclude the proof that Γ is a sphere. Let x be any point inside Γ, and y a point of Γ at minimal distance from x. The line L joining y to x is orthogonal to Γ at y, and meets Γ in another point z distinct from y , where again it is orthogonal to Γ, by (i). Let now w denote the midpoint of the chord joining y and z. We claim Γ is a sphere centered at w, with radius r equal to half the length of the chord joining y and z. Indeed, suppose there is a point of Γ at distance from w unequal to r, say greater than r. Then a point u exists on ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 77 Γ at maximal distance s from w, where s > r. A moment’s thought implies that u cannot be collinear with w and z, and the line joining w to u meets Γ orthogonally. Hence (ii) applies, and yields that w is equidistant from z and u, which is a contradiction. This concludes the proof.  Remarks. It may be of some interest to try to characterize those domains whose N - P kernel satisfies various weaker symmetry assump- tions, such as: a) K is symmetric ”modulo rank 1” , i.e. it is a symmetric kernel plus a ”perturbation” of the form a(x)b(y). b) The iterated (or, m times iterated) kernel associated to K is sym- metric. Yet another aspect of symmetrization of the kernel (in two dimen- sions) based on change of independent variable, was discussed in an interesting paper [10] by D. Gaier. 8.3. The ellipse. An analysis of the single and double layer potential operators on an ellipse goes back to Neumann [31]. The computations, also reproduced in the book by Plemelj [33], start with the elliptical 80 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO energy. As we have seen, this charge is in the kernel of the N - P oper- ator if and only if the normal derivatives of these functions on Γ (w.r.t. say the outer normal n) are everywhere negatives of each other, that is ∂nu+ (u 2 + v2)∂nu− u(2u∂nu+ 2v∂nv) = 0 on Γ, (39) or, simplifying (1− u2)∂nu− uv∂nv = 0 on Γ. Substituting 1− u2 = v2 and cancelling v, this becomes v∂nu = u∂nv on Γ. By virtue of the Cauchy - Riemann equations, ∂nu = ∂τv and ∂τu = −∂nv, where τ denotes the unit tangent vector to Γ. Thus, the last equation is equivalent to u∂τu+ v∂τv = 0 along Γ. But, this is true, it is just the result of differentiating u2 + v2 = 1 along Γ in the tangen- tial direction. Since all the steps are reversible, (39) is proved and the charge defined by the above potential is indeed in the kernel. Apply- ing the identical procedure, but starting in turn with the polynomials P 2, P 3, P 4, ... we get infinitely many elements in the kernel. These are linearly independent, since their potentials all have different rates of decay at ∞. The theorem is proved.  ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 81 Remark. The multiplication trick at the end to get the infinite di- mensionality seems to have no counterpart in more than 2 dimensions. Although the first part of the proof also was heavily dependent on two dimensional features (harmonic conjugates and Cauchy - Riemann) it does not seem beyond credibility that an analogous example could be found in 3 or more dimensions, i.e. an example where the kernel is nontrivial. More precisely, calculations analogous to those above lead to the conclusion that a sufficient condition for existence of a domain in R3 with non-injective N - P operator is the affirmative resolution of the following Hypothesis. There are three real polynomials p, q, r of 3 variables satisfying the following conditions: a) p is harmonic; b) p, q, r have no common zero on R3; c) s := p2 + q2 + r2 tends to ∞ at ∞; d) p/s is harmonic; e) Denoting by Ω a nonempty component of the set {s < 1} we have along the surface ∂Ω the identity ∂np p = ∂nq q = ∂nr r . 82 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO One could presumably write a computer program to search for such a triple among low degree polynomials. Note that already for the disk the closure of the single layer potentials which belong to kerK in the energy norm coincides with W 1/2 modulo constants, i.e. with the subspace in W 1/2 consisting of all functions on the circle with the mean value zero. This explains the painstaking caution one must obey in the statement of Theorem 4.2: unlike for eigenfunctions corresponding to nonzero eigenvalues one cannot really expect much additional regularity, e.g. membership in W 1/2, for mass distributions χ for which Sχ ∈ kerK. Furthermore, for general lemniscates we still do not know whether the closure of <P n,=P n, n = 1, 2, ... (cf. the notation in the proof of Theorem 8.3) covers all of kerK, i.e. whether the preimages of these functions with respect to the operator S are dense in the space of all distributions χ ∈ W−1/2 such that Sχ ∈ kerK ( cf. Theorem 4.2). Even less is known in higher dimensions. As we saw ( Section 8.1) in the ball kerK = 0. We do not know any particular example of a bounded domain in Rd with a nontrivial kerK, yet we strongly suspect that there are such domains. For unbounded domains the situation is completely different, e.g., for the half-space K is simply a trivial zero operator, so it is kernel is all of L2. ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 85 [5] J. Deny, Sur la définition de l’énergie en théorie du potentiel, Ann. Inst. Fourier 2(1950), 83-99. [6] N. Dunford, J.T. Schwartz, Linear Operators, Vol. I, II, Interscience, New York, 1958, 1963. [7] P. Ebenfelt, D. Khavinson, H.S. Shapiro, An inverse problem for the double layer potential, Comput. Methods Funct. Theory 1 (2001), 387–401. [8] P. Ebenfelt, D. Khavinson, H.S. Shapiro, A free boundary problem related to single-layer potentials, Ann. Acad. Sci. Fenn. Math. 27 (2002), 21–46. [9] K.O. Friedrichs, P.D. Lax, On symmetrizable differential operators, Proc. Sym- pos. Pure Math., Amer. Math. Soc. Providence, R.I., 10(1967), 128-137. [10] D. Gaier, Über die Symmetrisierbarkeit des Neumannschen Kerns, Z. Angew. Math. Mech. 42(1962), 569-570. [11] I.M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Dif- ferential Operators (in Russian), Gos. Iz. Fiz. Mat., Moscow, 1963. [12] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Amer. Math. Soc. Translations Vol. 18, Amer. Math. Soc., Providence, R.I., 1969. [13] N.M. Günther, Die Potentialtheorie und ihre Anwendung auf Grundaufgaben der Mathematischen Physik, Teubner, Leipzig, 1957. [14] E. Hellinger, O. Toeplitz, Integralgleichungen und Gleichungen mit vielen un- endlich Unbekanten, Encyklopädie der Mathematischen Wissenschaften- Anal- ysis, vol. II C 13, Teubner, Leipzig, 1909-1921; reprint in book format by Chelsea, New York, NY, 1953. 86 DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO [15] O.D. Kellogg, Foundations of Potential Theory, J. Springer, Berlin, 1929. [16] A. Korn, Lehrbuch der Potentialtheorie 2 vol., Dümmler Verlag, Berlin, 1899. [17] A. Korn, Eine Theorie der linearen Integralgleichungen mit unsymmetrischen kernen, Tohoku J. Math. 1(1911-1912), 159-186; part II, ibidem, 2(1912-1913), 117-136. [18] A. Korn , Über die erste und zweite Randwertaufgabe der Potentialtheorie, Rend. Circ. Matem. Palermo 35(1913), 317-323. [19] A. Korn, Über die Anwendung zur Lösung von lineare Integralgleichungen mit unsymmetrischen Kernen, Arch. Math. 25(1916), 148-173; part II, ibidem, 27(1918), 97-120. [20] M.G. Krein, Compact linear operators on functional spaces with two norms, (Ukrainian), Sbirnik Praz. Inst. Mat. Akad. Nauk Ukrainsk SSR 9(1947), 104- 129; English translation in: Integral Equations Oper. Theory 30:2(1998), 140- 162. [21] V.D. Kupradze (ed.), Three Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North Holland, Amsterdam, 1979. [22] T. Lalesco, Introduction à la Théorie des Équations Intégrales, Hermann, Paris, 1912. [23] T.Lalesco, Les classes de noyaux symmetrisables, Bull. Soc. Math. France 45(1917), 144-149. [24] N.S. Landkof, Foundations of Modern Potential Theory. Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer, Berlin, 1972. ON POINCARÉ’S VARIATIONAL PROBLEM IN POTENTIAL THEORY 87 [25] P. Lax, Symmetrizable linear transformations, Commun. Pure Appl. Math. 7(1956), 633-647. [26] V.B. Lidskii, On the eigenvalues of sums and products of symmetric matrices, Dokl. Akad. Nauk SSSR 75(1950), 769-772. [27] J.L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Ap- plications, Vol. I, Springer, Berlin, 1972. [28] J. Marty, Valeurs singulières d’une équation de Fredholm, C.R. Acad. Sci. Paris 150(1910), 1499-1502. [29] V.G. Maz’ya, Boundary integral equations, Analysis IV. Linear and boundary integral equations (V.G. Maz’ya and S. M. Nikol’skii, eds.) , Encycl. Math. Sci. vol. 27, Springer, Berlin, 1991, pp. 127-222. [30] J. Mercer, Symmetrisable functions and their expansion in terms of biorthog- onal functions, Proc. Royal Soc. (A) 97 (1920), 401-413. [31] C. Neumann, Über die Integration der partiellen Differentialgleichung ∂ 2Φ ∂x2 + ∂2Φ ∂y2 = 0, J. reine angew. Math. 59(1861), 335-366. [32] A.J. Pell, Applications of biorthogonal systems of functions to the theory of integral equations, Trans. Amer. Math. Soc. 12(1911), 165-180. [33] J. Plemelj, Potentialtheoretische Untersuchungen, Teubner, Leipzig, 1911. [34] H. Poincaré, La méthode de Neumann et le problème de Dirichlet, Acta Math. 20(1897), 59-152. [35] H. Poincaré, Théorie du Potentiel Newtonien, Carré et Náud, Paris, 1899.