Excess-height bound, with application to a harmonic measure free - PowerPoint PPT Presentation

Excess-height bound, with application to a harmonic measure free boundary problem Zihui Zhao Institute for Advanced Study joint work with S. Bortz, M. Engelstein, M. Goering, T. Toro Zihui Zhao 1 / 10

We say a set E ⊂ R n is a set of locally finite perimeter , if �� � div ϕ dX : ϕ ∈ C 1 c ( R n , R n ) and � ϕ � ∞ ≤ 1 sup < + ∞ . E It then follows that � � ϕ · ν E d H n − 1 , for any ϕ ∈ C 1 c ( R n , R n ) , div ϕ dX = − ∂ ∗ E E where ∂ ∗ E ⊂ ∂ E is called the reduced boundary of E , ν E the unit normal vector to E and | ν E | = 1 a.e.. We say the vector-valued Radon measure µ E := ν E H n − 1 | ∂ ∗ E is the Gauss-Green measure of E . Note that | µ E | = H n − 1 | ∂ ∗ E is the perimeter measure of E . Zihui Zhao 2 / 10

Proposition (Excess-height bound) Suppose E is a set of locally finite perimeter such that ∂ E = spt µ E and | µ E | is Ahlfors regular. There exists ǫ > 0 such that if the excess | ν E − e n | 2 1 � d H n − 1 < ǫ e n ( x 0 , 2 r ) := r n − 1 2 C ( x 0 , 2 r , e n ) ∩ ∂ ∗ E for x 0 ∈ ∂ E and r > 0 , then 1 1 2( n − 1) . r sup {| q ( x ) − q ( x 0 ) | : x ∈ C ( x 0 , r , e n ) ∩ ∂ E } ≤ Ce n ( x 0 , 2 r ) Moreover, there exists a Lipschitz function u : R n − 1 → R with Lip u ≤ 1 such that H n − 1 ( M ∆ Gr( u )) ≤ Ce n ( x 0 , r ) , where M := C ( x 0 , r , e n ) ∩ ∂ E . r n − 1 Zihui Zhao 3 / 10

Proposition (Excess-height bound) Suppose E is a set of locally finite perimeter such that ∂ E = spt µ E and | µ E | is Ahlfors regular. There exists ǫ > 0 such that if the excess | ν E − e n | 2 1 � d H n − 1 < ǫ e n ( x 0 , 2 r ) := r n − 1 2 C ( x 0 , 2 r , e n ) ∩ ∂ ∗ E for x 0 ∈ ∂ E and r > 0 , then 1 1 2( n − 1) . r sup {| q ( x ) − q ( x 0 ) | : x ∈ C ( x 0 , r , e n ) ∩ ∂ E } ≤ Ce n ( x 0 , 2 r ) Remark This type of excess-height bound has been known for a long time if the set E minimizes the perimeter. This bound was the first step towards proving the regularity of perimeter minimizers. (c.f. De Giorgi) Zihui Zhao 3 / 10

Proposition (Excess-height bound) Suppose E is a set of locally finite perimeter such that ∂ E = spt µ E and | µ E | is Ahlfors regular. There exists ǫ > 0 such that if the excess | ν E − e n | 2 1 � d H n − 1 < ǫ e n ( x 0 , 2 r ) := r n − 1 2 C ( x 0 , 2 r , e n ) ∩ ∂ ∗ E for x 0 ∈ ∂ E and r > 0 , then 1 1 2( n − 1) . r sup {| q ( x ) − q ( x 0 ) | : x ∈ C ( x 0 , r , e n ) ∩ ∂ E } ≤ Ce n ( x 0 , 2 r ) Remark Roughly speaking, it says if the unit normal of a domain Ω has small 1 n − 1 Reifenberg-flat; moreover oscillation (bounded by ǫ ), its boundary is ǫ Ω and Ω c is well-separated by the strip. Zihui Zhao 3 / 10

WLOG we may assume x 0 = 0 and r = 1. Lemma (Separation lemma) Given t 0 ∈ (0 , 1) , there exists δ = δ ( t 0 ) such that if e n (0 , 2) < δ , then | q ( x ) | < t 0 , for any x ∈ M , and |{ x ∈ C (0 , 1 , e n ) ∩ E : q ( x ) < − t 0 }| = 0 , |{ x ∈ C (0 , 1 , e n ) ∩ E c : q ( x ) > t 0 }| = 0 , Zihui Zhao 4 / 10

Proposition (Compactness for sets of finite perimeter) Suppose { E k } is a sequence of sets of locally finite perimeter whose boundaries are Ahlfors regular with uniform constants and ∂ E k = spt µ E k . Assume 0 ∈ ∂ E k , then passing to a subsequence there are a set of locally finite perimeter E and a Radon measure µ such that E k → E in L 1 , µ E k ⇀ µ E , | µ E k | ⇀ µ. Moreover, µ is Ahlfors regular, | µ E | ≤ µ , and 1 If x ∈ ∂ E, then there exist x k ∈ ∂ E k such that x k → x; 2 If x k ∈ ∂ E k and x k → x, then x ∈ spt µ . Zihui Zhao 5 / 10

Proof of the proposition Proof by pictures. Consider the function f : ( − 1 , 1) → [0 , H n − 1 ( M )] defined by f ( t ) = H n − 1 ( M ∩ { q ( x ) > t } ) . Recall that the excess bounds the difference between this measure and the measure of its projection onto R n − 1 . Zihui Zhao 6 / 10

Proof of the proposition Proof by pictures. Consider the function f : ( − 1 , 1) → [0 , H n − 1 ( M )] defined by f ( t ) = H n − 1 ( M ∩ { q ( x ) > t } ) . Recall that the excess bounds the difference between this measure and the measure of its projection onto R n − 1 . We denote the slicings E t = { z ∈ R n − 1 : ( z , t ) ∈ E } and ( ∂ ∗ E ) t = { z ∈ R n − 1 : ( z , t ) ∈ ∂ ∗ E } . By the co-area formula � � � � g d H n − 2 dt = 1 − ( ν E · e n ) 2 d H n − 1 g ( ∂ ∗ E ) t R ∂ ∗ E Zihui Zhao 6 / 10

Application to harmonic measure For any E ⊂ ∂ Ω, its harmonic measure is � � Brownian motion B X ω ( E ) := P t exits the domain Ω from E . Zihui Zhao 7 / 10

Application to harmonic measure Consider the Dirichlet boundary value problem for the Laplacian � − ∆ u = 0 , in Ω u = f , on ∂ Ω . The harmonic measure is the unique measure such that � f d ω X . u ( X ) = ∂ Ω Zihui Zhao 7 / 10

One expects that the harmonic measures for nice domains have good behavior, e.g. ω ≪ σ := H n − 1 | ∂ Ω . See the work of Dahlberg, David-Jerrison, Semmes, Hofmann-Martell, etc. Zihui Zhao 8 / 10

One expects that the harmonic measures for nice domains have good behavior, e.g. ω ≪ σ := H n − 1 | ∂ Ω . See the work of Dahlberg, David-Jerrison, Semmes, Hofmann-Martell, etc. The type of problem we consider is the converse: Harmonic measure free boundary problem Suppose the harmonic measure has nice behavior (for example, suppose log d ω d σ ∈ VMO), what can we deduce about the domain? For previous results in this direction, see Jerison, Kenig-Toro, Azzam-Tolsa-Mourgoglou etc. and Azzam-Hofmann-Martell-Mayboroda- Mourgoglou-Tolsa-Volberg. Zihui Zhao 8 / 10

All previous quantitative results require some a-priori assumptions on the domain (e.g. strong connectivity assumption), so that we have some PDE estimates to start the analysis. Zihui Zhao 9 / 10

All previous quantitative results require some a-priori assumptions on the domain (e.g. strong connectivity assumption), so that we have some PDE estimates to start the analysis. Our approach is: log d ω singular integral excess − height bound d σ ∈ VMO = = = = = = = = = ⇒ ν ∈ VMO = = = = = = = = = = = = ⇒ Ω is nice . ↑ ∇S f ( Z ) = 1 lim 2 ν ( x ) f ( x ) + T f ( x ) Z → x Z ∈ Γ( x ) Zihui Zhao 9 / 10

Thank you! Zihui Zhao 10 / 10