The Dirichlet Problem for Harmonic Functions on Compact Sets
aa r X i v : . [ m a t h . C A ] J a n THE DIRICHLET PROBLEM FOR HARMONIC FUNCTIONS ONCOMPACT SETS
TONY L. PERKINS
Abstract.
The primary goal of this paper is to study the Dirichlet problem ona compact set K ⊂ R n . Initially we consider the space H ( K ) of functions on K which can be uniformly approximated by functions harmonic in a neighborhoodof K as possible solutions. As in the classical theory, our Theorem 6.1 shows C ( ∂ f K ) ∼ = H ( K ) for compact sets with ∂ f K closed, where ∂ f K is the fineboundary of K . However, in general a continuous solution cannot be expectedeven for continuous data on ∂ f K as illustrated by Theorem 6.1. Consequently,we show that the solution can be found in a class of finely harmonic functions.Moreover by Theorem 6.5, in complete analogy with the classical situation,this class is isometrically isomorphic to C b ( ∂ f K ) for all compact sets K . Introduction
The Dirichlet problem for harmonic functions on domains in R n is not onlyimportant by itself but by its influence on potential theory. Many now standardnotions, e.g. regular points, fine topology, etc., first appeared in the study of thisproblem. The main goal of the present paper is to extend the classic theory tocompact sets K ⊂ R n .One possible extension can be found in the abstract theory of balayage spaces,see [BH86, H85]. However we feel that the gain in transparency following from adirect geometric approach more than justifies the use of new techniques.The Dirichlet problem can be thought of as having two components; the dataset and the data itself. One uses an initial function defined on the data set toconstruct a solution (a harmonic function) on the rest of the domain which musthave a prescribed regularity as it approaches the data set. Classically, the data setis taken to be the topological boundary of the domain. One of the main goals ofthis paper is to establish that the natural choice for the data set on compact setsis the fine boundary of K , ∂ f K , which is shown by Lemma 5.1 to be the Choquetboundary of K with respect to subharmonic functions on K . We limit ourselves toinitial functions that are continuous and bounded on ∂ f K as in the classical case.In Section 3, we introduce Jensen measures as our main tool and begin extend-ing potential theory to compact sets K ⊂ R n by defining harmonic functions andsubharmonic functions on K . We devote Section 4 to the construction and study ofharmonic measure on compact sets. The harmonic measure on K is shown to be amaximal Jensen measure. This is used to see the important fact (Corollary 5.3) thatharmonic measures are concentrated on the fine boundary. In Section 6 we study Date : November 21, 2018.2000
Mathematics Subject Classification.
Primary: 31B05; Secondary: 31B10, 31B25, 31C40.
Key words and phrases.
Harmonic measure, Jensen measures, Subharmonic functions, Poten-tial theory, Fine topology. he Dirichlet problem for compact sets. As in the classical theory, our Theorem 6.1shows C ( ∂ f K ) ∼ = H ( K ) for a class of compact sets. However, in general a continu-ous solution cannot be expected even for continuous data on ∂ f K as illustrated byExample 6.2. Consequently, we show that the solution can be found in the class offinely harmonic functions introduced in this section. Moreover by Theorem 6.5, incomplete analogy with the classical situation, this class is isometrically isomorphicto C b ( ∂ f K ) for all compact sets K .It is a pleasure to thank Stephen J. Gardiner, Leonid Kovalev, and GregoryVerchota for stimulating discussions related to the topic of this article. We areespecially grateful to Eugene Poletsky for his excellent guidance and support.2. Basic Facts
First some notation. Let M (Ω) denote the space of finite signed Radon measureson Ω ⊂ R n and C ( R n ) will be the space of continuous functions on R n which vanishat infinity. We will often use µ ( f ) to denote R f dµ .2.1. Classical Potential Theory.
Let D be an open set in R n , n ≥
2. Forany f ∈ C ( ∂D ), the Dirichlet problem on D is to find a unique function h whichis harmonic on D and continuous on D such that h | ∂D = f . The function f iscommonly referred to as the boundary data , and the corresponding h is said to bethe solution of the Dirichlet problem on D with boundary data f . The punctureddisk in R is a fundamental example which shows that the Dirichlet problem cannot be solved for any continuous boundary data. However for a bounded open set U the method of Perron allows one to assign a function which is harmonic on U to any continuous (or simply measurable) boundary data. Later the concept of aregular domain was developed to establish the continuity of the Perron solution tothe boundary. A bounded connected open set D ⊂ R n is a regular domain if theDirichlet problem is solvable on D for any continuous boundary data. Thereforeon a regular domain, C ( ∂D ) is isometrically isomorphic to H ( D ), the space ofcontinuous functions on D which are harmonic on D . For any f ∈ C ( ∂D ) let h f ∈ H ( D ) denote the solution of the Dirichlet problem on D with boundary data f . Let z ∈ D . The point evaluation H z : f h f ( z ) is a positive bounded linearfunctional on C ( ∂D ). By the Riesz Representation Theorem, there is a Radonmeasure ω D ( z, · ) on ∂D which represents H z , that is h f ( z ) = Z ∂D f ( ζ ) dω D ( z, ζ ) , for all f ∈ C ( ∂D ). The measure ω D ( z, · ) is called the harmonic measure of D withbarycenter at z . See [AG01] for more details on potential theory.2.2. Jensen Measures. If D is an open set in R n , we say that µ is a Jensenmeasure on D with barycenter z ∈ D if µ is a probability measure (a positiveRadon measure of unit mass) whose support is compactly contained in D and forevery subharmonic function f on D the sub-averaging inequality f ( z ) ≤ µ ( f ) holds.The set of Jensen measures on D with barycenter z ∈ D will be denoted J z ( D ).One could define the set of Jensen measures J cz ( D ) with respect to the continuoussubharmonic functions on D . However the following theorem shows that the set ofJensen measures would not be changed. heorem 2.1. Let D be a bounded open subset of R n . For every z ∈ D , the sets J z ( D ) and J cz ( D ) are equal.Proof. Since it is clear that J z ( D ) ⊆ J cz ( D ) for all z ∈ D , we will now show thereverse inclusion.Pick some z ∈ D and let µ ∈ J cz ( D ). Then we must show f ( z ) ≤ µ ( f )for every function f which is subharmonic on D . The support of µ is compactlycontained in D .Since f is subharmonic on D we can find an decreasing sequence { f n } of con-tinuous subharmonic functions which converge to f . As µ ∈ J cz ( D ) we have f ( z ) ≤ µ ( f n ) for every f n . By the Lebesgue Monotone Convergence Theorem itfollows that f ( z ) ≤ µ ( f ). Thus µ ∈ J z ( D ). (cid:3) Since J z ( D ) = J cz ( D ) for all z ∈ D , to check that µ ∈ J z ( D ), it suffices to checkthat µ has the sub-averaging property for every continuous subharmonic function.Examples of Jensen measures with barycenter at z ∈ D include the Dirac mea-sure at z , i.e. δ z , the harmonic measure with barycenter at z for any regulardomain which is compactly contained in D , and the average over any ball (orsphere) centered at z which is contained in D . The following proposition of Coleand Ransford [CR01, Proposition 2.1] will demonstrate some basic properties ofsets of Jensen measures. Proposition 2.2.
Let D and D be open subsets of R n with D ⊂ D . Let z ∈ D . i. If µ ∈ J z ( D ) then also µ ∈ J z ( D ) . ii. If µ ∈ J z ( D ) and supp( µ ) ⊂ D , and if each bounded component of R n \ D meets R n \ D , then µ ∈ J z ( D ) . Jensen measures and subharmonic functions are, in a sense, dual to each other.This duality is illustrated by the following theorem of Cole and Ransford [CR97,Corollary 1.7].
Theorem 2.3.
Let D be an open subset of R n which possesses a Green’s function.Let φ : D → [ −∞ , ∞ ) be a Borel measurable function which is locally bounded above.Then, for each z ∈ D , sup { v ( z ) : v ∈ S ( D ) , v ≤ φ } = inf { µ ( φ ) : µ ∈ J z ( D ) } , where S ( D ) denotes the set of subharmonic functions on D . Fine Topology.
The two books [B71, F72] are classical references on the finetopology and many books on potential theory contain chapters on the topic, e.g.[AG01, Chapter 7].The fine topology on R n is the coarsest topology on R n such that all subhar-monic functions are continuous in the extended sense of functions taking values in[ −∞ , ∞ ].When referring to a topological concept in the fine topology we will follow thestandard policy of either using the words “fine” or “finely” prior to the topologicalconcept or attaching the letter f to the associated symbol. For example, the fineboundary of K , ∂ f K , is the boundary of K in the fine topology. The fine topologyis strictly finer than the Euclidean topology.Many of the key concepts of classical potential theory have analogous definitionsin relation to the fine topology. Presently we will recall a few of them. Relative o a finely open set V in R n the harmonic measure δ V c x is defined as the swept-out of the Dirac measure δ x on the complement of V . A function u is said tobe finely hyperharmonic on a finely open set U if it is lower finite, finely lowersemicontinuous, and −∞ < δ V c x ( u ) ≤ u ( x ) , for all x ∈ V and all relatively compact finely open sets V with fine closure con-tained in U . A function h is said to be finely harmonic if h and − h are finelyhyperharmonic. Furthermore, the fine Dirichlet problem on U for a finely con-tinuous function f defined on the fine boundary of a bounded finely open set U consists of finding a finely harmonic extension of f to U . The development of thefine Dirichlet problem is quite similar to that of the classical. In the seventies Fu-glede [F72] establishes a Perron solution for the fine Dirichlet problem. His [F72,Theorem 14.6] shows that there exists a Perron solution H Uf which is finely har-monic on U for any numerical function f on ∂ f U which is δ ∂ f Ux integrable for every x ∈ U . Furthermore [F72, Theorem 14.6] provides us with the desired continuityat the boundary, i.e. that the fine limit of H Uf ( x ) tends to f ( y ) as x ∈ U goes to y for every finely “regular” boundary point y ∈ ∂ f U at which f is finely continuous.3. Harmonic and Subharmonic Functions on Compact Sets
We now begin our study of potential theory on compact sets. For compact setswhich are not connected, the Hausdorff property will allow us to reduce Dirichlettype problems on the compact set to solving separate problems on each connectedcomponent. Therefore in what follows we will work on compact sets K in R n whichneed not be connected, with the understanding that we can always separate theproblem by working the connected components of K individually.There are currently three equivalent ways to define harmonic and subharmonicfunctions on compact sets. Definition 3.1 (Exterior) . Let H ( K ) (or S ( K ) ) be the unform closures of all func-tions in C ( K ) which are restrictions of harmonic (resp. subharmonic) functionson a neighborhood of K . Definition 3.2 (Interior) . One can define H ( K ) and S ( K ) as the subspaces of C ( K ) consisting of functions which are finely harmonic (resp. finely superhar-monic) on the fine interior of K . The equivalence of these definitions of H ( K ) was shown in [DG74] and of S ( K )in [BH75, BH78].For the third definition of H ( K ) we must to extend the notion of Jensen measuresto compact sets. Definition 3.3.
We define the set of Jensen measures on K with barycenter at z ∈ K as the intersection of all the sets J z ( U ) , that is J z ( K ) = \ K ⊂ U J z ( U ) , where U is any open set containing K . Another definition of H ( K ) was introduced in [P97] using the notion of Jensenmeasures. efinition 3.4 (Via Jensen measures) . The set H ( K ) is the subspace of C ( K ) consisting of functions h such that h ( x ) = µ ( h ) for all µ ∈ J x ( K ) and x ∈ K . It was shown in [P97] that this definition is equivalent to the exterior definitionabove.Our first lemma shows that this last construction of Poletsky extends to subhar-monic functions in the ideal way.
Lemma 3.5.
A function is in S ( K ) if and only if it is continuous and satisfies thesubaveraging property with respect to every Jensen measure on K , that is S ( K ) = { f ∈ C ( K ) : f ( z ) ≤ µ ( f ) , for all µ ∈ J z ( K ) and every z ∈ K } . Proof.
We use the exterior definition of S ( K ) to show “ ⊆ ”. Let { f j } be a sequenceof subharmonic functions defined in a neighborhood of K such that { f j } is converg-ing uniformly to f . Then f j ( z ) ≤ µ ( f j ) for any µ ∈ J z ( K ). Since the convergenceis uniform we have f ( z ) ≤ µ ( f ).Now suppose that f is in the set on the right. The subaveraging conditionimplies that f is finely subharmonic, and by assumption f is continuous. Therefore f satisfies the interior definition of S ( K ). (cid:3) Recall the (exterior) definition of S ( K ) as the uniform limits of continuous func-tions subharmonic in neighborhoods of K . The following proposition shows thatthe defining sequence for any function in S ( K ) may be taken to be increasing. Thisresult is a simple consequence of a duality theorem of Edwards. Proposition 3.6.
Every function in S ( K ) is the limit of an increasing sequenceof continuous subharmonic functions defined on neighborhoods of K .Proof. Recall (see [G78, Theorem 1.2]and [CR97]) Edwards Theorem states: If p isa continuous function on K , then for all z ∈ K we have Ep ( z ) := sup { f ( z ) : f ∈ S ( K ) , f ≤ p } = inf { µ ( p ) : µ ∈ J z ( K ) } . From the proof of this theorem it follows that Ep is lower semicontinuous and is thelimit of an increasing sequence of continuous subharmonic functions on neighbor-hoods of K . The result follows by observing that p = Ep whenever p ∈ S ( K ). (cid:3) Harmonic Measure on a Compact Set
To use the exterior definition of H ( K ) we will commonly want to approximate K by a decreasing sequence of regular domains. A decreasing sequence of regulardomains { U j } is said to be converging to K if for every ε > j suchthat U j lies in the ε -neighborhood K ε of K when j ≥ j . Furthermore, we requirethat U j +1 is compactly contained in U j , i.e. U j +1 ⊂ U j , for all j . The existence ofsuch a sequence is provided by [H62, Prop 7.1].The next theorem will allow us to define a harmonic measure on K . For a de-creasing sequence of regular domains { U j } , we will let ω U j ( z, · ) denote the harmonicmeasure on U j with barycenter at z ∈ U j . Theorem 4.1. If { U j } is a sequence of regular domains converging to a compactset K ⊂ R n , then for every z ∈ K the harmonic measures ω U j ( z, · ) converge weak ∗ .Furthermore, this limit does not depend on the choice of the sequence of domains { U j } . roof. Since ω U j are measures of unit mass supported on a compact set in R n , byAlaoglu’s Theorem they must have a limit point. To show that this point is uniqueit suffices to show that for every z ∈ K the limit(1) lim j →∞ Z ∂U j u ( ζ ) dω U j ( z, ζ )exists for every u ∈ C ( U ).First, we show the limit in (1) exists when u is continuous and subharmonic in aneighborhood of K . The solution u j of the Dirichlet problem on U j with boundaryvalue u is equal to u j ( z ) = Z ∂U j u ( ζ ) dω U j ( z, ζ ) . Since u is subharmonic, we have u j ≥ u on U j . Then as u j +1 = u on ∂U j +1 and u j ≥ u = u j +1 on ∂U j +1 , the maximum principle for harmonic functions impliesthat u j ≥ u j +1 on U j +1 . Thus { u j } is a decreasing sequence on K and we see thatfor every z ∈ K the limit in (1) exists.If u ∈ C ( U ), then we may represent u as a difference of two C ( U ) functionswhich are subharmonic on U . By the argument above the limit in (1) exists.Since C ( U ) is dense in C ( U ) we see that the limit in (1) always exists. (cid:3) Definition 4.2.
We define the harmonic measure ω K ( z, · ) on a compact set K with z ∈ K as the weak ∗ limit of ω U j ( z, · ) chosen as above. To use this definition for the Dirichlet problem we must check that the supportof ω K ( z, · ) lies on the boundary of K . Actually in Section 5 we will be able to givemore specific information about ω K ( z, · ), see Corollary 5.3. Lemma 4.3.
The support of ω K ( z, · ) is contained in ∂K .Proof. Let W be a neighborhood of ∂K . Let { U j } be a sequence of domainsconverging to K and take a sequence z j ∈ ∂U j . Then there exists a subsequence { z j k } which must be converging to some z ∈ K . As z j ∈ ∂U j , then z j is not in K .Therefore the limit of z j k cannot be in the interior of K . Thus z is in ∂K ⊂ W .Consequently, there is j such that ∂U j ⊂ W for each j ≥ j ,Let x ∈ R n \ ∂K and take W to be a neighborhood of ∂K so that x W . Thereis an r > B ( x, r ) ∩ W = ∅ . Since ω U j ( z, · ) has support on ∂U j , which iscontained is W for large j , we have ω U j ( z, B ( x, r )) = 0. Since B ( x, r ) is open, thePortmanteau Theorem showslim inf j →∞ ω U j ( z, B ( x, r )) ≥ ω K ( z, B ( x, r )) . Hence ω K ( z, B ( x, r )) = 0 and x is not in the support of ω K ( z, · ). (cid:3) The following theorem brings our study back to the topic of Jensen measures.
Theorem 4.4.
The harmonic measure on K is a Jensen measure on K .Proof. Since ω K ( z, · ) is defined as the weak ∗ limit of probability measures, ω K ( z, · )is a probability measure.Recall that for z ∈ K we have defined J z ( K ) = ∩J z ( U ), where K ⊂ U . Howeverit is sufficient to see that J z ( K ) = ∩J z ( U J ) where { U j } is any sequence of domainsconverging to K . We will show ω K ( z, · ) ∈ J z ( U j ) for all j . ick some j . Then let f be a continuous subharmonic function on U j . Then f ( z ) ≤ Z ∂U l f ( ζ ) dω U l ( z, ζ ) , for all l > j . Then by taking the weak ∗ limit, we have that f ( z ) ≤ Z ∂K f ( ζ ) dω K ( z, ζ ) . Then ω K ( z, · ) satisfies the sub-averaging inequality for every continuous subhar-monic function on U j and ω K ( z, · ) is a probability measure with support containedin U j . Thus ω K ( z, · ) must be in J cz ( U j ), which is equal to J z ( U j ) by Theorem 2.1.Therefore ω K ( z, · ) ∈ J z ( K ). (cid:3) Following [G78, p. 16] a partial ordering on the set of Jensen measures is definedbelow. The notation J ( K ) is used to stand for the union of all Jensen measureson K , that is J ( K ) = [ z ∈ K J z ( K ) . Definition 4.5.
For µ, ν ∈ J ( K ) we say that µ (cid:23) ν if for every φ ∈ S ( K ) we have µ ( φ ) ≥ ν ( φ ) . Furthermore, a Jensen measure µ is maximal if there is no ν (cid:23) µ with ν = µ where ν ∈ J ( K ) . We start with a simple observation.
Lemma 4.6. If µ ∈ J z ( K ) and ν ∈ J z ( K ) with z = z then µ and ν are notcomparable.Proof. To see this simply recall that the coordinate functions π i are harmonic. As z = z they must differ in at least one coordinate, say the i th . Assume with outloss of generality that π i ( z ) > π i ( z ). Then µ ( π i ) > ν ( π i ). However − π i is alsoharmonic and so ν ( − π i ) > µ ( − π i ). Therefore µ and ν are not comparable and if µ (cid:23) ν then they have the common barycenter. (cid:3) We will now show that the harmonic measure is maximal with respect to thisordering. The maximality of harmonic measure proved below is the LittlewoodSubordination Principle (see [D70, Theorem 1.7]) when K is the closed unit ball inthe plane. Theorem 4.7.
For all z ∈ K , the measure ω K ( z, · ) is maximal in J ( K ) .Proof. By Lemma 4.6 it suffices to show that for any z ∈ K , ω K ( z, · ) is maximalin J z ( K ).Pick any z ∈ K . Now we will show that ω K ( z , · ) majorizes every measure µ ∈ J z ( K ). Consider a decreasing sequence of regular domains { U j } convergingto K . Take any φ ∈ S c ( K ). By Proposition 3.6 we may find a sequence φ j ∈ S c ( U j )increasing to φ . Furthermore we extend φ as ˜ φ ∈ C ( R n ) while keeping ˜ φ ≥ φ j forall j . Define harmonic functions Φ j on U j byΦ j ( x ) = Z ∂U j +1 φ j ( ζ ) dω U j +1 ( x, ζ ) . herefore as φ j is subharmonic, Φ j ≥ φ j on U j +1 , so Z ∂U j +1 φ j ( ζ ) dω U j +1 ( z , ζ ) = Φ j ( z ) = µ (Φ j ) ≥ µ ( φ j ) . As ˜ φ ≥ φ j , we have(2) Z ∂U j +1 ˜ φ ( ζ ) dω U j +1 ( z , ζ ) ≥ µ ( φ j ) , for all j . By taking weak ∗ limits, we have thatlim j →∞ Z ∂U j +1 ˜ φ ( ζ ) dω U j +1 ( z , ζ ) = Z ∂K φ ( ζ ) dω K ( z , ζ ) . The Lebesgue Monotone Convergence Theorem provideslim j →∞ µ ( φ j ) = µ ( φ ) . By taking the limit by j of (3) we see Z ∂K φ ( ζ ) dω K ( z , ζ ) ≥ µ ( φ ) . Therefore we have ω K ( z , · ) (cid:23) µ . If any ν ∈ J z ( K ) has the property ν (cid:23) µ , bythe antisymmetry property of partial orderings ν = µ . Thus the measure ω K ( z , · )is maximal in J z ( K ). (cid:3) The maximality of harmonic measures implies that they are trivial at the points z ∈ K such that J z ( K ) = { δ z } , which by Lemma 5.1 are precisely the fine boundarypoints. Corollary 4.8.
The harmonic measure ω K ( z , · ) = δ z if and only if J z ( K ) = { δ z } .Proof. Suppose ω K ( z , · ) = δ z . Consider the function ρ ( z ) = || z − z || ∈ S c ( K ).Then for any µ ∈ J z , by the maximality of ω K ( z , · ) we have0 = ρ ( z ) ≤ µ ( ρ ) ≤ Z ∂K ρ ( ζ ) dω K ( z , ζ ) = ρ ( z ) = 0 . As ρ ( z ) > z = 0 and as µ is a probability measure, we see that µ = δ z .Thus J z ( K ) = { δ z } .For the reverse implication we have already proved Theorem 4.4 that ω K ( z , · ) ∈J z ( K ). (cid:3) The Boundary
In the book [G78], Gamelin introduces a version of Choquet theory for conesof functions on compact sets. (Actually it applies to sets of functions which areslightly weaker than the cones we define.)Following his guidance we consider a set R of functions mapping a compact set K ⊂ R n to [ −∞ , ∞ ) with the following properties:i. R includes the constant functions,ii. if c ∈ R + and f ∈ R then cf ∈ R , ii. if f, g ∈ R then f + g ∈ R , andiv. R separates the points of K .One then considers a set of R -measures for z ∈ K defined as the set of probabilitymeasures µ on K such that f ( z ) ≤ µ ( f )for all f ∈ R .Naturally our model for R will be S ( K ). It then follows that when R = S ( K )the R -measures for z ∈ K are precisely J z ( K ). We now state some classic resultsfrom [G78] which we will need in the following sections.One can define the Choquet boundary of K with respect to S ( K ) as Ch S ( K ) K = { z ∈ K : J z ( K ) = { δ z }} . Many nice properties of the Choquet boundary are known. In particular, we willneed the following characterization, see also, for example, [BH86, VI.4.1] and [H85].
Lemma 5.1.
The Choquet boundary of K with respect to S ( K ) is the fine boundaryof K , i.e. Ch S ( K ) K = ∂ f K. Proof.
Since the fine topology is strictly finer than the Euclidean topology, anypoint in the interior of K will also be in the fine interior of K , and any point of R n \ K can be separated from K by an Euclidean (therefore fine) open set. Thereforethe fine boundary of K is contained in ∂K . The result follows immediately from[P97, Theorem 3.3] or [BH86, Proposition 3.1] which states that J z ( K ) = { δ z } ifand only if the complement of K is non-thin at z , that is z is a fine boundary pointof K . (cid:3) The set ∂ f K is also called the stable boundary of K . In fact the lemma showsthat Ch S ( K ) K is the finely regular boundary of the fine interior of K . For moredetails on finely regular boundary points and other related concepts, see [BH86,VII.5-7] and [H85].With this association, the result in [B71, p. 89] of Brelot about the stableboundary points of K shows that Ch S ( K ) K is dense in ∂K . Theorem 5.2.
The fine boundary of K (and therefore the Choquet boundary of K with respect to S ( K ) ) is dense in the topological boundary of K . In general the fine boundary is not closed, as Example 6.2 of the Section 6will show. So we cannot claim that it is the support of measures. Moreover, asTheorem 5.2 just showed the closure of O k is the boundary of K . In particular, itmay coincide with K for porous Swiss cheeses, see [G84, pg. 25-26].Recall that a measure µ ∈ M ( K ) is concentrated on a set E , if for every set F ⊂ K \ E , µ ( F ) = 0. A probability measure µ is concentrated on a set E ifand only if µ ( E ) = 1. From [G78, p. 19] we know that all maximal measuresare concentrated on Ch S ( K ) K = ∂ f K . With this observation, the next corollaryimmediately follows from Theorem 4.7 which stated that the harmonic measure ismaximal. Corollary 5.3.
For every z in K , the harmonic measure with barycenter at z isconcentrated on ∂ f K . . The Dirichlet Problem on Compact Sets
In the classical setting we know that any continuous function in the boundaryof a domain D ⊂ R n extends harmonically to D and continuously to D if and onlyif every point of the boundary is regular. For general compact sets in R n we havethe following result.From this result it also follows that the swept-out point mass at z onto K is just ω K ( z, · ). Theorem 6.1. If K is a compact set in R n then any function φ ∈ C ( ∂ f K ) extendsto a function in H ( K ) if and only if the set ∂ f K is closed. Moreover, the solutionis given by Φ( z ) = Z ∂ f K φ ( ζ ) dω K ( z, ζ ) z ∈ K and H ( K ) is isometrically isomorphic to C ( ∂ f K ) .Proof. Suppose that the set ∂ f K is closed. Consider a continuous function φ on ∂ f K . Let Φ( z ) = Z ∂ f K φ ( ζ ) dω K ( z, ζ ) z ∈ K. As ∂ f K is closed, by Theorem 5.2, we have ∂ f K = ∂K . Also as ω K ( z, · ) = δ z forevery z ∈ ∂ f K , we see that Φ = φ on ∂ f K .Let z j be a sequence in K converging to z ∈ ∂ f K . As z is in ∂ f K = Ch S ( K ) K ,so J z ( K ) = { δ z } . Since (see [G78, p. 3]) J ( K ) is weak ∗ compact, any sequenceof measures µ j ∈ J z j ( K ) must converge weak ∗ to δ z . In particular, ω U j ( z j , · ) isweak ∗ converging to δ z . Hence Φ( z j ) is converging to Φ( z ) = φ ( z ), and Φ iscontinuous at the boundary of K .As ∂ f K is closed, we have φ ∈ C ( ∂ f K ) = C ( ∂K ). We extend φ continuously as˜ φ ∈ C ( R n ), and then define the harmonic functions h j ( z ) = Z ∂U j ˜ φ ( ζ ) dω U j ( z, ζ ) . As ˜ φ is continuous and ω U j ( z, · ) converges weak ∗ to ω K ( z, · ),lim j →∞ h j ( z ) = lim j →∞ Z ∂U j ˜ φ ( ζ ) dω U j ( z, ζ ) = Z ∂K φ ( ζ ) dω K ( z, ζ ) = Φ( z ) . Therefore Φ is the pointwise limit of a sequence { h j } of functions harmonic ina neighborhood of K . Furthermore we can take the extension ˜ φ of φ in such away that the sequence { h j } is uniformly bounded. It now easily follows that Φ iscontinuous on the interior of K . Indeed, consider a point z in the interior of K .Then there exists a ball B centered at z contained in the interior of K . The h j are harmonic functions on B and converging pointwise to Φ. Thus Φ is continuouson B by the Harnack principle, and so Φ is continuous on K . Therefore we have acontinuous function Φ with representationΦ( z ) = Z ∂K φ ( ζ ) dω K ( z, ζ ) z ∈ K. ince Φ is continuous on K by [P97] to check that Φ ∈ H ( K ) all that remains isto show that Φ is averaging with respect to Jensen measures, i.e. the equivalenceof the external definition of H ( K ) and the definition by Jensen measures. So weneed to see that Φ( z ) = µ z (Φ) for every µ z ∈ J z ( K ) and for every z ∈ K . As h j isharmonic on U j , h j ( z ) = µ z ( h j ). However by the Lebesgue Dominated ConvergenceTheorem µ z (Φ) = lim j →∞ µ z ( h j ) = lim j →∞ h j ( z ) = Φ( z ) . Thus Φ ∈ H ( K ).For the converse, suppose ∂ f K is not closed. Then there is a point z ∈ ∂K \ ∂ f K .Since z is not in ∂ f K , by Corollary 4.8, ω K ( z , · ) is not trivial. Therefore we canfind a set E ⊂ ∂K such that ω K ( z , E ) > E in the complement of B ( z , r )for some r >
0. Consider a continuous function f on ∂K such that f = 1 on ∂K outside B ( z , r ) is 1 and f = 0 on B ( z , r/ ∩ ∂K . Then Z ∂K f ( ζ ) dω K ( z , ζ ) > ω K ( z , E ) z ∈ K. However f ( z ) = 0. Thus there can be no function in H ( K ) which agrees with f on the boundary of K . (cid:3) Example . The following set provides a simple example of a compact set K ⊂ R n , n ≥
3, in which the fine boundary is not closed. The set K is obtained from theclosed unit ball B ⊂ R n by deleting a sequence { B ( z n , r n ) } ∞ n =1 of open balls whosecenters and radii tend to zero. We take the centers to be z n = (2 − n , , . . . , ∈ R n and the radii 0 < r n < − n − . This example is analogous to the “road runner”example of Gamelin [G84, Figure 2, pg 52] and the Lebesgue spine [AG01, pg 187].By Theorem 6.1 one can not expect a continuous solution for the Dirichlet prob-lem on an arbitrary compact set even with continuous boundary data. Therefore atthis point we consider the following broader class of solutions with weaker continuityrequirement. Definition 6.3.
Let f H c ( K ) denote the class of finely continuous functions on K which are finely harmonic on the fine interior of K and continuous and boundedon ∂ f K . We have seen (the definition via Jensen measures) that H ( K ) consists of thefunctions in C ( K ) satisfying the averaging property with respect to J ( K ) and bythe interior definition of H ( K ) can also be seen as the C ( K ) functions which arefinely harmonic on the fine interior of K . Therefore in the definition of f H c ( K ) wehave maintained the finely harmonic requirement while requiring continuity onlyon the boundary ∂ f K (to match the boundary data). In fact Theorem 6.5 belowshows that the functions in f H c ( K ) also satisfy the averaging property with respectto J ( K ).Theorem 6.5 will show that the Dirichlet problem on compact sets K ⊂ R n issolvable in the class of functions f H c ( K ) for boundary data that is continuous andbounded on ∂ f K . The functions which are continuous and bounded on ∂ f K willbe denoted C b ( ∂ f K ). For this we will need the following [F72, Theorem 11.9] ofFuglede. heorem 6.4. The pointwise limit of a pointwise convergent sequence of finelyharmonic functions u m in U , a finely open subset of R n , is finely harmonic providedthat sup m | u m | is finely locally bounded in U . Theorem 6.5.
For every φ ∈ C b ( ∂ f K ) , i.e. continuous and bounded on ∂ f K ,there is a unique h φ ∈ f H c ( K ) equal to φ on ∂ f K . Moreover, h φ satisfies theaveraging property for J ( K ) and in particular h φ ( x ) = Z ∂ f K φ ( ζ ) dω K ( x, ζ ) , x ∈ K. Proof.
Let φ ∈ C b ( ∂ f K ) and for x ∈ ∂ f K define˜ φ ( x ) = lim sup y → x, y ∈ ∂ f K φ ( y ) . Since φ is continuous on ∂ f K , if x ∈ ∂ f K then ˜ φ ( x ) = φ ( x ). Furthermore, ˜ φ is uppersemicontinuous, and as such we may find a decreasing sequence of functions { φ k } which are continuous on ∂ f K and converge pointwise to ˜ φ . Then we extend the φ k to C ( R n ) as ˆ φ k . By taking ˜ φ k = min { ˆ φ , ˆ φ , · · · , ˆ φ k } we can make the extensionsbe decreasing. Consider a decreasing sequence of regular domains U j converging to K . Let u j, k be the solution of the Dirichlet problem on U j for ˜ φ k . As the measures ω U j ( x, · ) weak ∗ converge to ω K ( x, · ), we have that lim j u j, k = R ˜ φ k dω K := u k . Asthe ˜ φ k are decreasing, u k must also be decreasing. Indeed, we will let h φ = lim u k .Take any µ ∈ J ( K ). Then µ ∈ J z ( U j ) for all j and some z ∈ K . As u j, k is harmonic, we have µ ( u j, k ) = u j, k ( z ). However by the Lebesgue DominatedConvergence Theorem we have lim j µ ( u j, k ) = µ ( u k ), and so µ ( u k ) = u k ( z ). Sincethe sequence { u k } is decreasing pointwise to h φ we have that µ ( h φ ) = h φ ( z ) bythe Lebesgue Monotone Convergence Theorem. Thus h φ satisfies the averagingproperty on J ( K ). As ω K ( z, · ) ∈ J ( K ) for all z ∈ K we see that h φ ( z ) = Z ∂ f K h φ ( ζ ) ω K ( z, ζ ) . We will now show that h φ = φ on ∂ f K . For any x ∈ O k , we know ω K ( x, · ) = δ x ,and u k ( x ) = lim j →∞ u j, k ( x ) = Z ˜ φ k ( ζ ) dω K ( x, ζ ) = ˜ φ k ( x ) . Thus u k ( x ) = ˜ φ k ( x ) for all x ∈ ∂ f K , and so h φ ( x ) = lim k →∞ u k ( x ) = lim k →∞ ˜ φ k ( x ) = φ ( x ) , for all x ∈ ∂ f K .To see that h φ is finely harmonic we use Theorem 6.4. Observe that u k is thepointwise limits of the harmonic (and therefore finely harmonic) functions u j, k ,and the solution h φ is the pointwise limit of u k . From the construction of thesefunctions it is clear that they are bounded. (cid:3) Corollary 6.6.
The set C b ( ∂ f K ) is isometrically isomorphic to f H c ( K ) .Proof. The previous theorem establishes the homomorphism taking C b ( ∂ f K ) to f H c ( K ). Observe that h | ∂ f K ∈ C b ( ∂ f K ) for every h ∈ f H c ( K ). The uniqueness f the solution shows that h | ∂ f K extends as h . Furthermore, the isometry followsdirectly from the integral representation in the previous theorem. (cid:3) References [AG01] Armitage, D. H., Gardiner, S. J.,
Classical Potential Theory,
Springer-Verlag, London,2001[BH86] Bliedtner, J., Hansen, W.,
Potential Theory,
Springer-Verlag, 1986.[BH75] Bliedtner, J., Hansen, W.,
Simplicial Cones in Potential Theory,
Inventiones Mathe-maticae, Volume 29, (1975), 83–110[BH78] Bliedtner, J., Hansen, W.,
Simplicial Cones in Potential Theory II (ApproximationTheorems),
Inventiones Mathematicae, Volume 46, (1978), 255–275[B71] Brelot, M.,
On topologies and Boundaries in Potential Theory,
Lecture Notes in Math-ematics 175, Springer-Verlag, Berlin, 1971[CR01] Cole, B. J., Ransford, T. J.,
Jensen measures and harmonic measures,
Journal f¨ur diereine und angewandte Mathamatik, Volume 541, (2001), 29–53[CR97] Cole, B. J., Ransford, T. J.,
Subharmonicity without Upper Semicontinuity,
Journal ofFunctional Analysis, Volume 147, (1997), 420–442[C90] Conway, J. B.,
A Course in Functional Analysis,
Springer-Verlag, 1990[DG74] Debiard, A., Gaveau, B.,
Potential fin et alg´ebres de fonctions analytiques,
I.J. Funct.Anal., Volume 16, (1974), 289–304[D70] Duren, P. L.,
Theory of H p spaces, Pure and Applied Mathematics, Volume 38, AcademicPress, 1970[F72] Fuglede, B.,
Finely Harmonic Functions,
Lecture Notes in Mathematics 289, Springer-Verlag, 1972[G78] Gamelin, T. W.,
Uniform Algebras and Jensen Measures,
Cambridge University Press,1978[G84] Gamelin, T. W.,
Uniform Algebras,
AMS Chelsea Publishing Company, 1984[G95] Gardiner, S. J.,
Harmonic Approximation,
Cambridge University Press, 1995[GL93] Gauthier, P. M., Ladouceur, S.,
Uniform Approximation and Fine Potential Theory,
Journal of Approximation Theory, Volume 72, (1993), 138–140[H85] Hansen, W.,
Harmonic and superharmonic functions on compact sets.
Illinois J. Math.,Volume 29, Number 1, (1985), 103–107.[H62] Herv´e, R.-M.,
Recherches axiomatiques sur la th´eorie des fonctions surharmoniques etdu potentiel.
Ann. Inst. Fourier, 12, (1962), 415-517[P96] Poletsky, E. A.,
Analytic geometry on compacta in C n , Mathematische Zeitschrift, Vol-ume 222, (1996), 407–424[P97] Poletsky, E. A.,
Approximation by Harmonic Functions,
Transactions of the AMS, Vol-ume 349, Number 11, (1997), 4415–4427