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INVARIANCE OF GREEN EQUILIBRIUM MEASURE ON THE DOMAIN 1 1 STAMATISPOULIASIS 0 2 n a Abstract. WeprovethattheGreenequilibriummeasureandtheGreen J equilibrium energy of a compact set K relative to the domains D and 4 Ω are the same if and only if D is nearly equal to Ω, for a wide class 1 of compact sets K. Also, we prove that equality of Green equilibrium ] measures arises if and only if the one domain is related with a level set V of theGreen equilibrium potential of K relative to theother domain. C . 1. Introduction h t Consider the following inverse problem in classical potential theory: Sup- a m pose K is a compact set in a domain D. Do the equilibrium measure and [ the equilibrium energy of K relative to D characterize the domain D? More generally, if we know the energy and the measure on K, what can we con- 1 v clude about the ambient domain D? We proceed to a rigorous formulation 3 of the problem. 7 LetD beaGreenian opensubsetof Rn anddenotebyG (x,y) theGreen 8 D function of D. Also let K be a compact subset of D. The Green equilibrium 2 . energy of K relative to D is defined by 1 0 I(K,D) = inf G (x,y)dµ(x)dµ(y), 1 D µ 1 ZZ : where the infimum is taken over all unit Borel measures µ supported on K. v The Green capacity of K relative to D is the number i X 1 r CD(K)= . a I(K,D) WhenI(K,D) < +∞,theuniqueunitBorelmeasureµ forwhichtheabove K infimum is attained is the Green equilibrium measure and the function UD (x) = G (x,y)dµ (y) µK D K Z is the Green equilibrium potential of K relative to D. See e.g. [6, p. 174] or [2, p. 134]. We denote by C (E) the logarithmic (n = 2) or Newtonian 2 (n ≥ 3)capacity oftheBorelsetE. WhenE ⊂ D, theequalities C (E) = 0 D and C (E) = 0 are equivalent; [6, p. 174]. If two Borel sets A,B ⊂ Rn differ 2 Date: 28 June 2010. 1991 Mathematics Subject Classification. Primary: 31B15; Secondary: 31A15, 30C85. Key words and phrases. Green capacity,equilibrium measure, level sets. 1 2 STAMATISPOULIASIS only on a set of zero capacity (namely, C (A\B) = C (B \A) = 0), then 2 2 n.e. we say that A,B are nearly everywhere equal and write A=B. Nearly everywhere equal sets have the same potential theoretic behavior. Suppose that D is a domain and E is a compact subset of D \ K. If C (E) = 0, then the Green capacity and the Green equilibrium measure 2 of K relative to the open sets D and D \E are the same. That happens because the sets D and D\E are nearly everywhere equal. The following question arises naturally: DoesthereexistaGreeniandomainΩ,notnearlyeverywhere equal to D, such that the Green capacity and the Green equilibrium measure of K relative to D and Ω are the same? In general the answer is positive. However, we shall show that for a large class of compact sets (for example for compact sets with nonempty interior) the answer is negative. That is, given a compact set K with nonempty interior, theGreen equilibriummeasureµ andthepositive numberC (K) K D completely characterizes D from the point of view of potential theory. There is a second question: Suppose that the Green equilibrium measures of K relative to D and Ωare thesame. How are thesets D andΩ related? In that case, the boundary of the domain relative to which K has smaller Green energy is a level set of the Green equilibrium potential of K relative to the other domain. In our main result we give an answer to the above questions, for compact sets with nonempty interior. Theorem 1. Let K be a compact subset of Rn with nonempty interior. Let D ,D be two Greenian subdomains of Rn that contain K and let µ and µ 1 2 1 2 be the Green equilibrium measures of K relative to D and D , respectively. 1 2 Then n.e. (i) µ = µ and I(K,D ) = I(K,D ) if and only if D =D . 1 2 1 2 1 2 (ii) If I(K,D )< I(K,D ) and the set 1 2 D˜ = {x ∈ D :UD2(x) > I(K,D )−I(K,D )} 2 2 µ2 2 1 contains K, we have µ = µ if and only if D n=.e.D˜ . 1 2 1 2 Moreover, we shall show that Theorem 1 is valid for a much wider class of compact sets (see Remark 4.2). In the following section we introduce the concepts of Green potential theory that are needed for our results. Theorem 1 is proved in section 3. In section 4 we examine the case of compact sets K with empty interior, we give some counterexamples and we pose a conjecture and a question. INVARIANCE OF GREEN EQUILIBRIUM MEASURE 3 2. Background Material WedenotebyB(x,r)andS(x,r)theopenballandthespherewithcenter xandradiusrinRn,respectively. ForasetE ⊂ Rn,theinterior,theclosure and the boundary of E are denoted by E◦, E and ∂E, respectively. The surface area of the unit sphere S(0,1) of Rn is denoted by σ . n If a property holds for all the points of a set A apart from a set of zero capacity we will say that the property holds for nearly every (n.e.) point of A. The logarithmic (n = 2) or Newtonian (n ≥ 3) kernel is denoted by log 1 , n = 2, K(x,y) := |x−y| 1 , n ≥ 3, ( |x−y|n−2 for (x,y) ∈ Rn×Rn. If D ⊂ Rn is a Greenian open set then G (x,y) = K(x,y)+H(x,y) D where, for fixed y, H(·,y) is the greatest harmonic minorant of K(·,y) on D. If µ is a measure with compact support on D, the Green potential of µ is the function UD(x) = G (x,y)dµ(y), x ∈ D. µ D Z Then (see [2, pp. 96-104]) UD is superharmonic on D, harmonic on D \ µ supp(µ) and the Riesz measure of UD is proportional to µ; more precisely, µ ∆UD = −κ µ, where ∆ is the distributional Laplacian on D and µ n σ , n= 2, κ = 2 n (n−2)σ , n≥ 3. n (cid:26) We shall need the following result for the boundary behavior of a Green potential. Theorem 2.1. [2, p. 148] Let Ω be a Greenian open subset of Rn and let µ be a Borel measure with compact support in Ω. Then lim UΩ(x) = 0, µ Ω∋x→ξ for nearly every point ξ ∈ ∂Ω. Also,ifUD istheGreenequilibriumpotentialofacompactsetK relative µK to D, then the equality (1) UD (x) = I(K,D) µK holds for all x ∈ K◦ and for nearly every point x ∈K; ([5, p. 138]). The following theorem gives a geometric interpretation of the fact that sets of zero capacity are negligible. Theorem 2.2. ([2, p. 125]). Let Ω ⊂ Rn be a domain and E a relatively closed subset of Ω with C (E) = 0. Then the set Ω\E is connected. 2 4 STAMATISPOULIASIS Remark 2.3. WhentheopensetD\K isconnected,itiscalledacondenser and the sets Rn \D and K are called the plates of the condenser; see [3, 1]. It is well known that Green potential theory and condenser theory are equivalent; see e.g. [4, p. 700-701] or [7, p. 393]. Therefore, Theorem 1 can be restated as a theorem about condensers. 3. Dependence of Green equilibrium measure on the domain Let D be a Greenian open subset of Rn and K a compact subset of D. First we shall examine the open sets bounded by the level surfaces of the Green equilibrium potential of K relative to D. The Green equilibrium measure is the same for each of these open sets. Lemma 3.1. Let D be a Greenian open subset of Rn and let K be a compact subset of D that has finite Green equilibrium energy relative to D. Also let µ be the Green equilibrium measure of K relative to D and for 0 < α < K I(K,D), let D = {x ∈ D : UD (x) > α}. α µK If K ⊂ D and µ is the Green equilibrium measure of K relative to D , α α α then µ = µ and α K I(K,D ) = I(K,D)−α. α Proof. TheGreen equilibriumpotential UD is a superharmonicfunction on µK D so, in particular, it is lower semicontinuous on D. Therefore the sets D = {x ∈ D :UD (x) > α} α µK are open for all α ∈ (0,I(K,D)). Choose α ∈ (0,I(K,D)) such that K ⊂ D . Consider the function U : α D 7→ R with α U(x) =UD (x)−α. µK Then U is harmonic on D \K. Also, by property (1) of the Green equi- α librium potentials, U has boundary values (I(K,D)−α) n.e. on ∂K and 0 on ∂D . Therefore, by the extended maximum principle and the boundary α behavior of the Green equilibrium potential UDα on D \K, µα α I(K,D ) UDα(x) = α U(x). µα I(K,D)−α Applying the distributional Laplacian on D we get α I(K,D ) ∆(UDα)= ∆ α (UD −α) , µα I(K,D)−α µK (cid:16) (cid:17) so I(K,D ) α −κ µ = −κ µ n α n K I(K,D)−α and I(K,D ) α µ = µ . α K I(K,D)−α INVARIANCE OF GREEN EQUILIBRIUM MEASURE 5 Sinceµ andµ areunitmeasures,weconcludethatµ = µ andI(K,D ) = α K α K α I(K,D)−α. (cid:3) In order to proof Theorem 1 we shall need some more lemmas. In the following lemma we show that the difference of two Green potentials of the same measure is a harmonic function. Lemma 3.2. Let D and D be two Greenian open subsets of Rn such that 1 2 D ∩D 6= ∅. Also, letµ be aBorel measure with compact support inD ∩D 1 2 1 2 such that the Green potentials UD1 and UD2 are finite on D ∩D . Then µ µ 1 2 the difference UD1 −UD2 µ µ is a harmonic function on D ∩D . 1 2 Proof. Let G (x,y) = K(x,y)+H (x,y) i i be the Green function of D , i = 1,2, respectively. Then i UD1(x)−UD2(x) = K(x,y)dµ(y)+ H (x,y)dµ(y) µ µ 1 ZK ZK − K(x,y)dµ(y)− H (x,y)dµ(y) 2 ZK ZK = [H (x,y)−H (x,y)]dµ(y). 1 2 ZK The difference H (x,y) − H (x,y) is a harmonic function on D ∩ D on 1 2 1 2 both variables x and y, separately. Therefore, x 7→ UD1(x)−UD2(x) = [H (x,y)−H (x,y)]dµ(y) µ µ 1 2 ZK is a harmonic function on D ∩D ; see ([5, Lemma 6.7, p. 103]). (cid:3) 1 2 We shall need the fact that particular parts of the boundary of an open set belong to the reduced kernel [6, p. 164] of the boundary. Lemma 3.3. Let Ω be an open subset of Rn. Suppose that Rn\Ω 6= ∅ and let O be a connected component of Rn\Ω. Then for all ξ ∈ ∂O and for all r > 0, C (B(ξ,r)∩∂O) > 0. 2 Proof. Let ξ ∈ ∂O and r > 0. Then B(ξ,r)\∂O intersects Ω and O, so it is not connected. Then, since B(ξ,r)∩∂O is a relatively closed subset of B(ξ,r), it follows from Theorem 2.2 that C (B(ξ,r)∩∂O) > 0. 2 (cid:3) The next lemma gives a characterization of nearly everywhere equal sets: the Green potential of any measure with compact support is the same. 6 STAMATISPOULIASIS Lemma3.4. LetD andD betwoGreeniandomainsofRn. ThenD n=.e.D 1 2 1 2 if and only if there exists a Borel measure µ with compact support in D ∩D 1 2 suchthat UD1,UD2 are nonconstant functionson each connected component µ µ of D ∩D and UD1 = UD2 on an open ball B ⊂ D ∩D . 1 2 µ µ 1 2 n.e. Proof. Suppose that D =D . Then G (x,y) = G (x,y) for all x,y ∈ 1 2 D1 D2 D ∩D ([2, Corollary 5.2.5, p. 128]). Therefore 1 2 UD1(x) = G (x,y)dµ(y) = G (x,y)dµ(y) = UD2(x), µ D1 D2 µ Z Z for all x ∈ D ∩D and for all Borel measures µ with compact support in 1 2 D ∩D . 1 2 Conversely, suppose that µ is a Borel measure with compact support in D ∩D such that UD1,UD2 are non constant functions on each connected 1 2 µ µ component of D ∩D and UD1 = UD2 on a ball B ⊂ D ∩D . By Lemma 1 2 µ µ 1 2 3.2, the function u= UD1 −UD2 µ µ is harmonic on D ∩ D and vanishes on the ball B. Therefore, by the 1 2 identity principle ([2, Lemma 1.8.3, p. 27]), u vanishes on the connected component A of D ∩D that contains B. That is, UD1 =UD2 on A. 1 2 µ µ Suppose that C (D \ A) > 0. We shall show that C (D ∩ ∂A) > 0. 2 1 2 1 Consider the decomposition D \A= (D ∩∂A)∪(D ∩(Rn\A)). 1 1 1 If D ∩(Rn\A) = ∅, then 1 C (D ∩∂A) = C (D \A) > 0. 2 1 2 1 Supposethat D ∩(Rn\A) 6= ∅. Let O bea connected component of Rn\A 1 that intersects D . Since D intersects O and Rn \O, D ∩∂O 6= ∅. Let 1 1 1 ξ ∈ D ∩∂O and r > 0 such that B(ξ,r) ⊂D . By Lemma 3.3, 1 1 C (D ∩∂A) ≥ C (B(ξ,r)∩∂O) > 0. 2 1 2 Therefore, in any case C (D ∩∂A) > 0. Then, since ∂A ⊂ (∂D ∪∂D ) 2 1 1 2 and D ∩∂A ⊂ D , we have that D ∩∂A is a subset of ∂D with positive 1 1 1 2 capacity. From Theorem 2.1 we have that there exist ξ ∈ D ∩∂A such 0 1 that lim UD2(x) = lim UD2(x) = 0. µ µ A∋x→ξ0 D2∋x→ξ0 Since supp(µ) is a compact subset of D ∩D , ξ ∈ D and ξ ∈ ∂D , UD1 1 2 0 1 0 2 µ is harmonic on an open neighborhood of ξ . So 0 lim UD1(x) = lim UD1(x) = UD1(ξ ) > 0. µ µ µ 0 A∋x→ξ0 D1∋x→ξ0 Then, since UD1 = UD2 on A, we obtain µ µ 0 = lim UD2(x) = lim UD1(x)> 0, µ µ A∋x→ξ0 A∋x→ξ0 INVARIANCE OF GREEN EQUILIBRIUM MEASURE 7 which is a contradiction. n.e. Therefore C (D \A) = 0. Since A ⊂ D , we get D =A. In a similar 2 1 1 1 way we can show that D n=.e.A. Therefore, in particular, D n=.e.D . (cid:3) 2 1 2 We proceed to prove Theorem 1. n.e. Proof of Theorem 1. (i) Suppose that D =D . Then the relations 1 2 µ = µ and I(K,D )= I(K,D ) follow from the equality 1 2 1 2 G (x,y) = G (x,y) D1 D2 for all x,y ∈ K ([2, Corollary 5.2.5, p. 128]). Conversely, suppose that µ = µ and I(K,D )= I(K,D ). By property 1 2 1 2 (1) of the Green equilibrium potentials, UD1(x)−UD2(x) = I(K,D )−I(K,D ) = 0, µ1 µ2 1 2 for all x ∈ K◦. Then, by Lemma 3.4, D n=.e.D . 1 2 (ii) Let µ˜ be the Green equilibrium measure of K relative to D˜ . By 2 2 Lemma 3.1, µ˜ = µ and I(K,D˜ ) = I(K,D ). Suppose that D n=.e.D˜ . 2 2 2 1 1 2 Since D is a domain, by Theorem 2.2, D˜ is also a domain. Then by (i), 1 2 µ = µ˜ = µ . 1 2 2 Conversely, supposethatµ = µ . LetO beaconnectedcomponentofD˜ 1 2 2 thatintersectstheinteriorofK. Fromproperty(1)oftheGreenequilibrium potentials, UD1(x)−UD˜2(x) = I(K,D )−I(K,D˜ ) = 0, µ1 µ˜2 1 2 for all x ∈ K◦∩O. Also, UD˜2 = UO on O and µ = µ = µ˜. By Lemma µ˜2 µ˜2|O 1 2 2 n.e. 3.4, D =O. Therefore 1 D ∩(D˜ \O) = ∅ 1 2 and K ⊂ O, since K ⊂ D˜ . Suppose that D˜ 6= O and let A be a second 2 2 connected component of D˜ . Then K ∩A = ∅, UD2 is harmonic on A and 2 µ2 UD2 = I(K,D )−I(K,D ) µ2 2 1 on ∂A. By the maximum principle, UD2 = I(K,D ) − I(K,D ) on A µ2 2 1 and by the identity principle UD2 = I(K,D )−I(K,D ) on the connected µ2 2 1 component B of D \K that contains A. Also ∂B ⊂ (∂D ∪∂K). But 2 2 lim UD2(x)= 0 6= I(K,D )−I(K,D ), for n.e. ζ ∈ ∂D µ2 2 1 2 D2∋x→ζ and UD2 = I(K,D )6= I(K,D )−I(K,D ), n.e. on ∂K, µ2 2 2 1 which is a contradiction. Therefore D˜ = O and D n=.e.D˜ . 2 1 2 Remark 3.5. Let B be a ball that contains K. If the open set B \K is regular for the Dirichlet problem, the property K ⊂ D˜ is always true. 2 8 STAMATISPOULIASIS 4. Compact sets with empty interior and counterexamples Let K be a compact subset of Rn and let D ,D be two Greenian subdo- 1 2 mains of Rn that contain K. We shall examine the case where the domains D ,D arenotnearly everywhereequalandtheGreen equilibriummeasures 1 2 of K relative to D ,D are the same. It turns out that, in most cases, K 1 2 must be sufficiently smooth. Theorem 4.1. Let K be a compact subset of Rn with positive capacity and let D ,D be two Greenian subdomains of Rn that are not nearly everywhere 1 2 equalandcontainK. Alsoletµ andµ betheGreenequilibriummeasuresof 1 2 K relative to D and D , respectively. If µ = µ and I(K,D ) = I(K,D ), 1 2 1 2 1 2 then there exists a level set L of a non constant harmonic function such that C (K \L)= 0. 2 Proof. By Lemma 3.2, the function h(x) = UD1(x)−UD2(x) µ1 µ2 is harmonic on D ∩ D . Since D ,D are not nearly everywhere equal, 1 2 1 2 Lemma3.4showsthathcannotbe0onanon-emptyopensubsetofD ∩D . 1 2 From property (1) of the Green equilibrium potentials, h(x) = UD1(x)−UD2(x) =I(K,D )−I(K,D ) = 0, µ1 µ2 1 2 for nearly every point x ∈ K. So, h is a non constant harmonic function on every connected component A of D ∩D such that C (A∩K) > 0. Let 1 2 2 G be the union of all the connected components A of D ∩ D such that 1 2 C (A∩K)> 0 and let 2 L = {x ∈ G :h(x) =0}. Then h is a non constant harmonic function on G, L is a level set of h and C (K \L)= 0. (cid:3) 2 Remark 4.2. It follows from Theorem 4.1 that Theorem 1 is valid for all compact subsets K that cannot be contained, except on a set of zero capacity, on a level set of a non constant harmonic function. An example of a compact set in the above class is every (n − 1)-dimensional compact submanifold of Rn that is not real analytic on a relatively open subset. Another example is every compact set K ⊂ R2 which has at least one connected component that is neither a singleton nor a piecewise analytic arc. We proceedtogive examplesof compactsets K andpairsofdomainsthat are answers to the first question we posed in the introduction. Keeping in mind Remark 4.2, the compact sets will lie on sufficiently smooth subsets of Rn, mainly spheres and hyperplanes. Let K be a compact subset of a sphere S(x ,r) with C (K) > 0. Let 0 2 D be a Greenian open subset of Rn that contains K. We denote by D∗ INVARIANCE OF GREEN EQUILIBRIUM MEASURE 9 the inverse of D with respect to S(x ,r); see e.g. [2, p. 19]. Then D∗ is 0 Greenian and (see e.g. [2, p. 95]) r2 n−2 GD∗(x,y) = GD(x∗,y∗), x,y ∈ D∗. |x−x | |y−x | 0 0 (cid:16) (cid:17) Moreover, GD(x,y) = GD∗(x,y) for every x,y ∈ K. So I(K,D) = I(K,D∗) andtheGreenequilibriummeasuresofK relativetoD andD∗ arethesame. Of course, the sets D and D∗ are not nearly everywhere equal in general. A similar result holds also in the case when K is a subset of a hyperplane H and D∗ is the reflection of D with respect to H. Finally we state a conjecture and a question: Conjecture: Let K be a subset of a sphere S(x ,r) with C (K) > 0 and 0 2 let D,Ω be two Greenian domains that contain K. If I(K,D) = I(K,Ω) and the Green equilibrium measures of K relative to D and Ω are the same, thenΩn=.e.D or Ωn=.e.D∗ whereD∗ is theinverseof D withrespecttoS(x ,r). 0 Question: Letn ≥ 3. LetK beacompactsubsetofRn andsupposethat theredoesnotexistasphereS orahyperplaneH suchthatC (K\S) = 0or 2 C (K\H) = 0. DothereexistdomainsD,Ωwhicharenotnearlyeverywhere 2 equalandcontainK suchthatI(K,D) = I(K,Ω)andtheGreenequilibrium measures of K relative to D and Ω are the same? References [1] G.D.Anderson,M.K.Vamanamurthy,TheNewtoniancapacityofaspacecondenser, Indiana Univ.Math. J. 34 (1985), 753–776. [2] D.H.ArmitageandS.J.Gardiner,ClassicalPotential Theory,SpringerMonographs in Mathematics, Springer, 2001. [3] T. Bagby, The modulus of a plane condenser, J. Math. Mech. 17 (1967), 315–329. [4] M.Götz,Approximating the condenser equilibrium distribution,Math.Z.236(2001), no. 4, 699–715. [5] L. L. Helms, Introduction to Potential Theory, Wiley-Interscience, 1969. [6] N. S. Landkof,Foundations of Modern Potential Theory, Springer-Verlag, 1972. [7] E.B.SaffandV.Totik,LogarithmicPotentialswithExternalFields,Grundlehrender MathematischenWissenschaften[FundamentalPrinciplesofMathematicalSciences], 316, Springer-Verlag, 1997. Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece E-mail address: [email protected]