Condenser capacity under holomorphic functions

Transcription

1 under holomorphic functions Faculty of Engineering and Natural Sciences, Sabanci University Istanbul Analysis Seminars 12 February 2016 under holomorphic functions

2 Definition A condenser in C, is a pair (D, K), where D is an open subset of C and K is a nonempty compact subset of D. Definition u Lip(C) if there exists C > 0 such that Definition u(x) u(y) C x y. Ψ(D, K) is the family of functions u Lip(C) such that u = 1 on K and u = 0 on C \ D. The capacity of (D, K) is { } Cap(D, K) = inf u 2 : u Ψ(D, K). D\K under holomorphic functions

3 Definition The equilibrium potential h of the condenser (D, K) is the solution of the Dirichlet problem on D \ K with boundary values { 0, on D, φ = 1, on K. From the Dirichlet Principle, Cap(D, K) = D\K h 2. under holomorphic functions

4 under holomorphic functions

5 Examples If 0 < r < s < + and B(0, t) = {x C : x < 1}, then Cap(B(0, s), B(0, r)) = 1 log s log r under holomorphic functions

6 Definition The Green function G D (, y) : D (0, + ], with pole at y D, is harmonic on D \ {y}, 1 x G D (x, y) log x y, is harmonic on D, lim x ζ G D (x, y) = 0, for every ζ D except on a subset of zero capacity Examples D = {x C : x < 1}, G D (x, y) = log 1 xȳ, x, y D. x y under holomorphic functions

7 Green function of a rectangle with pole at 0. under holomorphic functions

8 Definition The Green energy of a unit Borel measure µ on K is I D (µ) = G D (x, y)dµ(x)dµ(y) K K and the Green potential of µ is Uµ D (x) = G D (x, y)dµ(y), x D. U D µ is harmonic on D \ K and lim x ζ UD µ (x) = 0, for every ζ D except on a subset of zero capacity. under holomorphic functions

9 Definition The Green equilibrium energy of the condenser (D, K) is I (D, K) = inf µ I D(µ). under holomorphic functions

10 Definition If I (D, K) < +, there is a unique unit Borel measure µ K on K, such that I D (µ K ) = I (D, K) = inf µ I D(µ). µ K is the Green equilibrium measure of (D, K). For the Green potential of µ K it is true that U D µ K I (D, K), on D, and U D µ K = I (D, K), on K except on a subset of zero capacity. under holomorphic functions

11 Then, and h(x) = UD µ K (x) I (D, K), G D (x, y)dµ K (x)dµ K (y) = 1 2π Cap(D, K) = 2π I (D, K). D U D µ K 2 under holomorphic functions

12 Theorem (Subordination Principle for Green function) Let f be a holomorphic function on the Greenian domain D C and suppose that f (D) is Greenian. Then G D (z, w) G f (D) (f (z), f (w)) Equality holds for two distinct points z 0, w 0 D if and only if f is injective. under holomorphic functions

13 Theorem (Conformal invariance of condenser capacity) If f is a conformal mapping on D, Cap(f (D), f (K)) = Cap(D, K). Theorem (Subordination Principle for condenser capacity) If f is a holomorphic function on D, Cap(f (D), f (K)) Cap(D, K). Equality holds if and only if f is univalent. under holomorphic functions

14 Let f : D C be a holomorphic function. n(x) is the multiplicity of f at x D v(u) = f (a)=u n(a) is the valency of f at u f (D). Theorem (Lindelöf Principle) Let f be a holomorphic function on the Greenian domain D C and suppose that f (D) is Greenian. Then G f (D) (u 0, f (x)) where x D and u 0 f (D). f (a)=u 0 n(a)g D (a, x), under holomorphic functions

15 Theorem (Equality statement, M. Heins (1955)) If there exists u 0 f (D) with v(u 0 ) = p < + such that G f (D) (u 0, f (x)) = n(a)g D (a, x), x D, f (a)=u 0 then v(u) = p for every u f (D) except on a set of zero capacity. under holomorphic functions

16 Suppose that (D, K) is a condenser, f is a holomorphic function on D, the condenser (f (D), f (K)) has positive capacity, ν is the Green equilibrium measure of (f (D), f (K)). under holomorphic functions

17 Let E := supp(ν), R f (u, K) := {x K : f (x) = u}, u E, N f (u, K) the cardinality of the set R f (u, K), u E, p = min u E N f (u, K). under holomorphic functions

18 Theorem (with M. Papadimitrakis) Equality 1 2πp 2 1 Cap(D, K) 1 p p i=1 f (a)=u a D\L 1 Cap(f (D), f (K)) n(a)g D (a, f 1 i (v))dν(u)dν(v). Cap(f (D), f (K)) = 1 Cap(D, K) p holds if and only if v(u) = p for every u f (D) except on a set of zero capacity. under holomorphic functions

19 Proof. From the theory of Borel sets we obtain that there exist Borel sets L i K, i = 1, 2,..., p, such that L i L j =, for every i j, the restriction f i of f to L i is an injective Borel measurable function, f (L i ) = E. L := p L i. i=1 under holomorphic functions

20 We will pull back the measure ν on K. Let and µ = 1 p µ i (E) = ν(f i (E)), E L i p µ i. i=1 Then µ is a unit Borel measure on K. under holomorphic functions

21 I (D, K) G D (x, y)dµ(x)dµ(y) K K = 1 p p 2 G D (x, y)dµ i (x)dµ j (y) i,j=1 L i L j = 1 p p 2 G D (f 1 i (u), f 1 j (v))dν(u)dν(v) i,j=1 = 1 p p 2 j=1 E E p E E i=1 G D (f 1 i (u), f 1 j (v))dν(u)dν(v) under holomorphic functions

22 1 p p 2 j=1 f (a)=u a D\L E E = 1 I (f (D), f (K)) p p 1 p 2 j=1 [ G f (D) (u, v) ] n(a)g D (a, f 1 j (v)) dν(u)dν(v) E E f (a)=u a D\L n(a)g D (a, f 1 j (v))dν(u)dν(v). under holomorphic functions

23 Suppose that Cap(f (D), f (K)) = 1 Cap(D, K). p Then G f (D) (u, f (z)) = p i=1 G D (f 1 i (u), z), z D, ν almost every point u E. Therefore v(u) = p for every u f (D) except on a set of zero capacity. under holomorphic functions

24 Let v(u) = p for every u f (D) except on a set of zero capacity. If h and h f are the equilibrium potentials of (D, K) and (f (D), f (K)), we obtain h = h f f. under holomorphic functions

25 From the nonunivalent change of variables formula, Cap(D, K) = h(x) 2 dm(x) = = = D\K D\K D\K (h f f )(x) 2 dm(x) h f (f (x)) 2 f (x) 2 dm(x) f (D)\f (K) p h f (u) 2 dm(u) = p Cap(f (D), f (K)). under holomorphic functions

26 Let B : D D be a Blaschke product with infinitely many zeroes, Let If for every n N B(z) = z n z n z z n 1 z n z. n=1 R n = {z D : 1 2 n z < 1 2 n 1 }. (B 1 (0) R n ) M counting multiplicities for a M N, B is called an exponential Blaschke product. under holomorphic functions

27 Let (D, C) be a condenser with positive capacity and let K n = B 1 (C) {z D : z 1 2 n }. Then How fast? lim Cap(D, K n) = +. n + under holomorphic functions

28 Theorem (with J. Mashreghi) Cap(D, K n ) = O(n), as n +. More precisely, there M = M(B, C) > 0 such that lim sup n + Cap(D, K n ) n MCap(D, C). under holomorphic functions

29 If every point of C has at least one pre-image in R n, then V B (K n ) n and 0 < Cap(D, C) Cap(D, K n ) lim inf n + V B (K n ) Cap(D, K n ) lim sup n + n MCap(D, C). Therefore, in general, we cannot replace O(n) by O(n t ) for some t < 1. under holomorphic functions

30 Let E be a compact subset of D with zero capacity which contains at least two points and let B : D D \ E be a universal covering map of D \ E. Then B is an (interpolating) Blaschke product that is not an exponential Blaschke product such that for every p N, Cap(D, K n ) lim n + n p = +. under holomorphic functions

31 Let D be a multiply connected Greenian domain in the complex plane and let f : D D be a universal covering map of D. Let u D, let V D be a fundamental neighborhood of u and let E = D(u, ρ) V. Let E 1, E 2,..., be an enumeration of the connected components of f 1 (E) and let n K n = E i. i=1 under holomorphic functions

32 Theorem (with J. Mashreghi) Then Cap(D, E) Cap(D, K n) n (Note that Cap(D, E i ) = Cap(D, E j ).) If D is doubly connected, then Cap(D, E 1 ), n N. Cap(D, K n ) lim = Cap(D, E). n n under holomorphic functions

33 Sketch of proof when D is doubly connected. Since V f (K n ) = n, or Cap(D, E) Cap(D, K n) n ni (D, K n ) I (D, E). under holomorphic functions

34 Lemma Let f n be the restriction of f to E n, n N. Then for every ɛ > 0 there exist m 0 N such that for every n N sup { n+m 0 G D (f (z), u) G D (z, f 1 i (u)) : z K n, u E } ɛ. i=1 under holomorphic functions

35 For every n N, let µ n be the Green equilibrium measure of (D, K n ) and consider the measure ν n (A) = µ n (f 1 (A)), A E Borel measurable. Then ν n is a probability Borel measure with support in E. Therefore I (D, E) G D (v, u)dν n (v)dν n (u) n N. under holomorphic functions

36 Fix ɛ > 0. I (D, E) lim inf n = lim inf n lim inf n = lim inf n = lim inf n G D (v, u)dν n (v)dν n (u) G D (f (z), u)dµ n (z)dν n (u) ( n+m 0 n+m0 i=1 n+m0 i=1 i=1 ) G D (z, f 1 i (u)) + ɛ dµ n (z)dν n (u) G D (z, f 1 i (u))dµ n (z)dν n (u) + ɛ Uµ D n (f 1 i (u))dν n (u) + ɛ under holomorphic functions

37 lim inf n n+m0 i=1 I (D, K n )dν n (u) + ɛ = lim inf (n + m 0)I (D, K n ) + ɛ n = lim inf ni (D, K n) + lim m 0I (D, K n ) + ɛ n n = lim inf ni (D, K n) + ɛ n lim sup ni (D, K n ) + ɛ I (D, E) + ɛ. n under holomorphic functions

38 Therefore or lim ni (D, K n) = I (D, E) n Cap(D, K n ) lim = Cap(D, E). n n under holomorphic functions

39 Teşekkür ederim! under holomorphic functions