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Harmonic Measure from Two Sides(and Tools from Geometric Measure Theory)

Matthew Badger

Department of MathematicsStony Brook University

April 12, 2012

Simons Postdoctoral Fellows Meeting

Research Partially Supported by NSF Grants DMS-0838212 and DMS-0856687

Dirichlet Problem

Let n 2 and let Rn be a domain.

Dirichlet Problem

(D)

{u = 0 in u = f on

= x1x1 +x2x2 + +xnxn

! family of probability measures {X}X on the boundary called harmonic measure of with pole at X such that

u(X ) =

f (Q)dX (Q) is the solution of (D)

Dirichlet Problem

Let n 2 and let Rn be a domain.

Dirichlet Problem

(D)

{u = 0 in u = f on

= x1x1 +x2x2 + +xnxn

! family of probability measures {X}X on the boundary called harmonic measure of with pole at X such that

u(X ) =

f (Q)dX (Q) is the solution of (D)

Dirichlet Problem

Let n 2 and let Rn be a domain.

Dirichlet Problem

(D)

{u = 0 in u = f on

= x1x1 +x2x2 + +xnxn

! family of probability measures {X}X on the boundary called harmonic measure of with pole at X such that

u(X ) =

f (Q)dX (Q) is the solution of (D)

Free Boundary Problem 1

Let Rn be a domain of locally finite perimeter, withharmonic measure and surface measure = Hn1|.

If the Poisson kerneld

dis sufficiently regular,

then how regular is the boundary ?

FBP 1 Results

I (Kinderlehrer and Nirenberg 1977) Let Rn be of class C 1.

1. log dd C1+m, for m 0, (0, 1) = is C 2+m,.

2. log dd C = is C

3. log dd is real analytic = is real analytic.

I (Alt and Caffarelli 1981) Assume Rn satisfies necessaryweak conditions (that includes C 1 as a special case). Then:log dd C

0, for > 0 = is C 1,, = () > 0.

I (Jerison 1987) In Alt and Caffarellis Theorem, = .

I (Jerison 1987) log dd C0 = is VMO1.

I (Kenig and Toro 2003) Studied FBP 1 with log dd VMO.

Examples of NTA Domains

Smooth Domains Lipschitz Domains Quasispheres

(e.g. snowflake)

Question: How should we measure regularity of harmonic measureon domains which do not have surface measure?

Free Boundary Problem 2

Rn is a 2-sided domain if:1. + = is open and connected

2. = Rn \ is open and connected3. + =

Let Rn be a 2-sided domain, equipped with interiorharmonic measure + and exterior harmonic measure

If the two-sided kerneld

d+is sufficiently regular,

then how regular is the boundary ?

An Unexpected Example

log d

d+ is smooth does not imply is smooth

Figure: The zero set of the harmonic polynomialh(x , y , z) = x2(y z) + y 2(z x) + z2(x y) 10xyz

= {h > 0} is a 2-sided domain, + = (pole at infinity),log d

d+ 0 but = {h = 0} is not smooth at the origin.

Structure Theorem for FBP 2

Theorem (B) Assume Rn is a 2-sided NTA domain,+ + and log dd+ VMO(d

+) or log d

d+ C0().

There exists d 1 (depending on the NTA constants) such that = 1 2 d .

1. Every blow-up of about a point Q k is the zero seth1(0) of a homogeneous harmonic polynomial h of degree kwhich separates Rn into two components.

2. The flat points 1 is a dense open subset of withHausdorff dimension n 1.

3. The singularities 2 d have harmonic measure zero.

Ingredients in the Proof

1. FBP 2 was studied by Kenig and Toro (2006) who showedthat blow-ups of are zero sets of harmonic polynomials.

I We show that only zero sets of homogeneous harmonicpolynomials appear as blow-ups.

I We show the degree of polynomials appearing in blow-ups isunique at every Q . Hence = 1 2 d .

I We study topology and size of the sets k .

2. To classify geometric blow-ups of the boundary, we studymeasure-theoretic blow-ups of (tangent measures).

3. To show 1 is open, we study local flatness properties of thezero sets of harmonic polynomials.

Polynomial Harmonic Measures

h : Rn R be a polynomial, h = 0

+ = {X : h(X ) > 0}, = {X : h(X ) < 0}(i.e. h is the Green function for )

The harmonic measure h associated to h is the harmonic measureof with pole at infinity; i.e., for all Cc (Rn),

h1(0)dh =

h

d =

h

Two Collections of Measures Associated to Polynomials

Pd = {h : h harmonic polynomial of degree d}Fk = {h : h homogenous harmonic polynomial of degree = k}

Blow-ups of the Boundary Tangent Measures of

Let Rn be a 2-sided NTA domain, let Q and let ri 0.

Theorem: (KT) There is subsequence of ri (which we relabel) andan unbounded 2-sided NTA domain such that

I Blow-ups of Boundary at Q Converge:

i = Q

ri in Hausdorff metric

I Blow-ups of Harmonic Measure at Q Converge:

i (E ) =(Q + riE )

(B(Q, ri ))satisfy i

where is the harmonic measure of with pole at infinity.

Each blow-up is called a tangent measure of at Q.

Tangent Measures of when + +

Theorem (Kenig and Toro)

If + + and log dd+ VMO(d+),

then Tan(,Q) Pd .

Goal: Show Tan(,Q) Fkfor some 1 k = k(Q) d .

Pd = {h : h harmonic polynomial of degree d}Fk = {h : h homogenous harmonic of degree = k}

Tangent Measures of when + +

Theorem (Kenig and Toro)

If + + and log dd+ VMO(d+),

then Tan(,Q) Pd .

Goal: Show Tan(,Q) Fkfor some 1 k = k(Q) d .

Pd = {h : h harmonic polynomial of degree d}Fk = {h : h homogenous harmonic of degree = k}

Cones of Measures

A collection M of non-zero Radon measures is a d-cone if itpreserved under scaling and dilation of Rn:

1. If M and c > 0, then c M.2. If M and r > 0, then T0,r] M. [T0,r (y) = y/r ]

Examples

Tangent Measures: Tan(, x)

Polynomial Harmonic Measures: Pd and Fk

Size of a Measure and Distance to a Cone

Let be Radon measure on Rn. The size of on B(0, r) isFr () =

r0 (B(0, s))ds.

Let be a Radon measure on Rn and M a d-cone. There is adistance dr (,M) from to M on B(0, r) compatiblewith weak convergence of measures.

Connectedness of Tangent MeasuresLet F and M be d-cones such that F M. Assume that:

F and M have compact bases ({ : F1() = 1}), (Property P) There exists 0 > 0 such that whenever M and dr (,F) < 0 for all r r0 then F .

TheoremIf Tan(, x) M and Tan(, x) F 6= , then Tan(, x) F .

Key Point: (Under technical hypotheses) If one tangent measureat a point belongs to F then all tangent measures belong to F .

First proved in [P], the theorem was stated in this form in [KPT].

Preiss used the theorem to show Radon measures in Rn withpositive and finite m-density almost everywhere are m-rectifiable.

Kenig, Preiss and Toro used the theorem to compute Hausdorffdimension of harmonic measure when + +.

Checking the Hypotheses: Rate of Doubling

I If Fk , then (B(0, r)) = crn+k2 where c depends on n,k and hL1(Sn1). Thus Fk is uniformly doubling: if Fkthen

(B(0, 2r))

(B(0, r))= 2n+k2 for all r > 0

independent of the associated polynomial h.Lemma: Fk has compact basis for all k 1.

I If Pd is associated to a polynomial of degree j d(not necessarily homogeneous), then for all > 1

(B(0, r))

(B(0, r)) n+j2 as r .

Theorem: The comparison constant depends only on n and j!

Corollary: If dr (,Fk) < 0(n, d) r r0(), then k = j .

Checking the Hypotheses: Rate of Doubling

I If Fk , then (B(0, r)) = crn+k2 where c depends on n,k and hL1(Sn1). Thus Fk is uniformly doubling: if Fkthen

(B(0, 2r))

(B(0, r))= 2n+k2 for all r > 0

independent of the associated polynomial h.Lemma: Fk has compact basis for all k 1.

I If Pd is associated to a polynomial of degree j d(not necessarily homogeneous), then for all > 1

(B(0, r))

(B(0, r)) n+j2 as r .

Theorem: The comparison constant depends only on n and j!

Corollary: If dr (,Fk) < 0(n, d) r r0(), then k = j .

Polynomial Blow-ups are Homogeneous

Theorem (B)

Let be a 2-sided NTA domain. If Tan(+,Q) Pd , thenTan(,Q) Fk for some 1 k d.

Steps in the Proof

1. Since Tan(+,Q) Pd , there is a smallest degree k dsuch that Tan(+,Q) Pk 6= . Show thatTan(+,Q) Pk Fk .

2. Let F = Fk and M = Tan(+,Q) Fk . By the previousslide the hypotheses of the connectedness theorem aresatisfied. Therefore, Tan(+,Q) Fk .

Open Questions

log d

d+ VMO(d+) = 1 2 d .

1. Find an upper bound on dimension of the singularities2 d . (Conjecture: dimH n 3)

2. (Higher Regularity) For example, if log d

d+ C0,, then at

Q k is locally the C 1, image of the zero set of aharmonic polynomial of degree k?

3. (Rectifiability) Does 1 = G N where G is (n 1)-rectifiableand (N) = 0?

I The answer is yes if one assumes that has locally finiteperimeter (Kenig-Preiss-Toro, B).

4. Find other applications of the connectedness of tangentmeasures.

REFERENCES

M. Badger, Harmonic polynomials and tangent measures ofharmonic measure, Rev. Mat. Iberoam. 27 (2011), no. 3,841870. arXiv:0910.2591

M. Badger, Flat points in zero sets of harmonic polynomials andharmonic measure from two sides, preprint. arXiv:1109.1427

APPENDIX

Distance from Measure to a Con

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