Harmonic Functions on Compact Sets in R n

Transcription

1 Harmonic Functions on Compact Sets in R n Tony Perkins toperkin@syr.edu Syracuse University Department of Mathematics 215 Carnegie Building Syracuse, NY June 2009

2 A Classical Result Let B be a ball in R n. Then the space of functions harmonic on the ball and continuous on the closed ball are isometrically isomorphic with the space of continuous functions on the boundary of the ball h(b) = C( B). Furthermore Note that the map h(b) C( B) is simply the restriction of the harmonic function to the boundary. toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

3 Let K R n be a connected compact set. Definition We say a decreasing sequence of smooth domains U j is converging to K if for every ε > 0 there is a j 0 such that U j lies in the ε-neighborhood K ε of K when j j 0. Definition We define the Jensen measures on K with barycenter at x K as the intersection of the Jensen measures on U j, that is J x (K) = J x (U j ). toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

4 Harmonic functions on K, h(k) Definition We define the set h(k) to be the uniform limits of functions harmonic in a neighborhood of K. Furthermore It was shown by Poletsky that { h(k) = h C(K): h(x) = } h dµ, µ J x (K) and x K toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

5 Harmonic Measure on K Let ω j (z, ) denote the harmonic measure on U j with barycenter at z. Theorem If {U j } is a decreasing sequence of domains converging to a connected compact set K R n, then for every z K the harmonic measures ω j (z, ) converge weak-. Furthermore, this limit does not depend on the choice of the sequence of domains U j Definition We define the harmonic measure ω K (z, ) on a compact set K and z K as the weak- limit of ω j (z, ) chosen as above. toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

6 Integral representation of harmonic functions Theorem Let h C(K). Then h h(k) if and only if it has the representation h(z) = h(ζ) dω K (z, ζ) for all z in K. K toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

7 Peak points Definition A point z 0 K is a harmonic peak point, if there exists h h(k) such that h(z 0 ) = 1, h = 1 and h(y) < 1 for y K, y z 0. The function h is said to peak at z 0. We will denote the set of harmonic peak points on K as O K. toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

8 Properties of O K The set O K is G δ toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

9 Properties of O K The set O K is G δ x O K iff J x (K) = {δ x } toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

10 Properties of O K The set O K is G δ x O K iff J x (K) = {δ x } (Bishop - De Leeuw) for every x K there is a representing measure µ for h(k) at x such that µ(o K ) = 1. toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

11 Properties of O K The set O K is G δ x O K iff J x (K) = {δ x } (Bishop - De Leeuw) for every x K there is a representing measure µ for h(k) at x such that µ(o K ) = 1. (Poletsky) x O K iff R n \ K is non-thin at x. toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

12 Properties of O K The set O K is G δ x O K iff J x (K) = {δ x } (Bishop - De Leeuw) for every x K there is a representing measure µ for h(k) at x such that µ(o K ) = 1. (Poletsky) x O K iff R n \ K is non-thin at x. (Doob) Let Ω be a connected Greenian open set, then the set of points of Ω at which Ω is thin forms a set of zero harmonic measure for Ω. toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

13 Properties of O K The set O K is G δ x O K iff J x (K) = {δ x } (Bishop - De Leeuw) for every x K there is a representing measure µ for h(k) at x such that µ(o K ) = 1. (Poletsky) x O K iff R n \ K is non-thin at x. (Doob) Let Ω be a connected Greenian open set, then the set of points of Ω at which Ω is thin forms a set of zero harmonic measure for Ω. toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

14 Properties of O K The set O K is G δ x O K iff J x (K) = {δ x } (Bishop - De Leeuw) for every x K there is a representing measure µ for h(k) at x such that µ(o K ) = 1. (Poletsky) x O K iff R n \ K is non-thin at x. (Doob) Let Ω be a connected Greenian open set, then the set of points of Ω at which Ω is thin forms a set of zero harmonic measure for Ω. Theorem The set O K is dense in K. toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

15 An Analog of Bishop-De Leeuw & A Dual of Doob Theorem For all x K, ω K (x, O K ) = 1. toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

16 An Analog of Bishop-De Leeuw & A Dual of Doob Theorem For all x K, ω K (x, O K ) = 1. (Bishop - De Leeuw) for every x K there is a representing measure µ for h(k) at x such that µ(o K ) = 1. toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

17 An Analog of Bishop-De Leeuw & A Dual of Doob Theorem For all x K, ω K (x, O K ) = 1. (Bishop - De Leeuw) for every x K there is a representing measure µ for h(k) at x such that µ(o K ) = 1. (Poletsky) x O K iff R n \ K is non-thin at x. toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

18 An Analog of Bishop-De Leeuw & A Dual of Doob Theorem For all x K, ω K (x, O K ) = 1. (Bishop - De Leeuw) for every x K there is a representing measure µ for h(k) at x such that µ(o K ) = 1. (Poletsky) x O K iff R n \ K is non-thin at x. (Doob) Let Ω be a connected Greenian open set, then the set of points of Ω at which Ω is thin forms a set of zero harmonic measure for Ω. toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

19 Theorem If O K is closed, then the Dirichlet problem is solvable on K, the solution is given by Φ(x) = φ(ζ) dω K (x, ζ), and h(k) is isometrically isomorphic to C(O K ). toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

20 Simple Swiss Cheese Let B be the closed unit disk in the plane. Let x n 0 +, and B n = B(x n, r n ). Define K = B \ B n. If the r n are decreasing rapidly enough, then K has a non-trivial Jensen measure at the origin. toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

21 Quasi-harmonic Definition Let K be a connected compact set in the plane. Then a function u : K [, ) is quasi-harmonic if (a) u is bounded and Suslin, (b) for each x K and each Jensen measure µ for x, we have u(x) = u dµ. toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

22 The class qh(k) Definition We define qh(k) as the set of all quasi-harmonic functions on K which are continuous and bounded on O K. Furthermore If O K is closed, then qh(k) = h(k). toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

23 Theorem For every φ C b (O K ) there is a unique h φ qh(k) equal to φ on O K. Moreover, h φ (x) = φ(ζ) dω K (x, ζ) O K and h φ = φ. Thus qh(k) is isometrically isomorphic to C b (O K ). toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15

24 Bishop, E., De Leeuw, K., The representations of linear functionals by measures on sets of extreme points, Annales de l institut Fourier, Volume 9, (1959), Cole, B. J., Ransford, T. J., Subharmonicity without Upper Semicontinuity, Journal of Functional Analysis, Volume 147, (1997), Doob, J. L., Classical Potential Theory and Its Probabilistic Counterpart, Volume 262, Springer-Verlag, New York, 1984 Poletsky, E. A., Analytic Geometry on Compacta in C n, Mathematische Zeitschrift, Volume 222, (1996), Poletsky, E. A., Approximation by Harmonic Functions, Transactions of the AMS, Volume 349, Number 11, (1997), toperkin@syr.edu (SU) Harmonic Functions on Compact Sets in R n 23 June / 15