Two-phase free boundary problems for harmonic measure with H older - PowerPoint PPT Presentation

Two-phase free boundary problems for harmonic measure with H¨ older data (and blowups in multi-phase problems) Matthew Badger University of Connecticut April 21, 2018 Research partially supported by NSF DMS 1500382 and NSF DMS 1650546.

Dirichlet Problem and Harmonic Measure Let n ≥ 2 and let Ω ⊂ R n be a regular domain for (D). Dirichlet Problem Given f ∈ C c ( ∂ Ω), find u ∈ C 2 (Ω) ∩ C (Ω): � ∆ u = 0 in Ω (D) u = f on ∂ Ω ∆ = ∂ x 1 x 1 + ∂ x 2 x 2 + · · · + ∂ x n x n ∃ ! family of probability measures { ω X } X ∈ Ω on the boundary ∂ Ω called harmonic measure of Ω with pole at X ∈ Ω such that � f ( Q ) d ω X ( Q ) u ( X ) = solves (D) ∂ Ω For unbounded domains, we may also consider harmonic measure with pole at infinity.

Dirichlet Problem and Harmonic Measure Let n ≥ 2 and let Ω ⊂ R n be a regular domain for (D). Dirichlet Problem Given f ∈ C c ( ∂ Ω), find u ∈ C 2 (Ω) ∩ C (Ω): � ∆ u = 0 in Ω (D) u = f on ∂ Ω ∆ = ∂ x 1 x 1 + ∂ x 2 x 2 + · · · + ∂ x n x n ∃ ! family of probability measures { ω X } X ∈ Ω on the boundary ∂ Ω called harmonic measure of Ω with pole at X ∈ Ω such that � f ( Q ) d ω X ( Q ) u ( X ) = solves (D) ∂ Ω For unbounded domains, we may also consider harmonic measure with pole at infinity.

Dirichlet Problem and Harmonic Measure Let n ≥ 2 and let Ω ⊂ R n be a regular domain for (D). Dirichlet Problem Given f ∈ C c ( ∂ Ω), find u ∈ C 2 (Ω) ∩ C (Ω): � ∆ u = 0 in Ω (D) u = f on ∂ Ω ∆ = ∂ x 1 x 1 + ∂ x 2 x 2 + · · · + ∂ x n x n ∃ ! family of probability measures { ω X } X ∈ Ω on the boundary ∂ Ω called harmonic measure of Ω with pole at X ∈ Ω such that � f ( Q ) d ω X ( Q ) u ( X ) = solves (D) ∂ Ω For unbounded domains, we may also consider harmonic measure with pole at infinity.

Examples of Regular Domains NTA domains introduced by Jerison and Kenig 1982: Quantitative Openness + Quantitative Path Connectedness Smooth Domains Lipschitz Domains Quasispheres (e.g. snowflake)

Two-Phase Free Boundary Regularity Problem Ω ⊂ R n is a 2-sided domain if: 1 Ω + = Ω is open and connected 2 Ω − = R n \ Ω is open and connected 3 ∂ Ω + = ∂ Ω − Let Ω ⊂ R n be a 2-sided domain, equipped with interior harmonic measure ω + and exterior harmonic measure ω − . If ω + ≪ ω − ≪ ω + , then f = d ω − d ω + exists, f ∈ L 1 ( d ω + ) . Determine the extent to which existence or regularity of f controls the geometry or regularity of the boundary ∂ Ω .

Two-Phase Free Boundary Regularity Problem Ω ⊂ R n is a 2-sided domain if: 1 Ω + = Ω is open and connected 2 Ω − = R n \ Ω is open and connected 3 ∂ Ω + = ∂ Ω − Let Ω ⊂ R n be a 2-sided domain, equipped with interior harmonic measure ω + and exterior harmonic measure ω − . If ω + ≪ ω − ≪ ω + , then f = d ω − d ω + exists, f ∈ L 1 ( d ω + ) . Determine the extent to which existence or regularity of f controls the geometry or regularity of the boundary ∂ Ω .

Regularity of a boundary can be expressed in terms of geometric blowups of the boundary

Existence of Measure-Theoretic Tangents at Typical Points Theorem (Azzam-Mourgoglou-Tolsa-Volberg 2016) Let Ω ⊂ R n be a 2-sided domain equipped with harmonic measures ω ± on Ω ± . If ω + ≪ ω − ≪ ω + , then ∂ Ω = G ∪ N, where 1 ω ± ( N ) = 0 and H n − 1 G is locally finite, 2 ω ± G ≪ H n − 1 G ≪ ω ± G, 3 up to a ω ± -null set, G is contained in a countably union of graphs of Lipschitz functions f i : V i → V ⊥ i , V ∈ G ( n , n − 1) . In contemporary Geometric Measure Theory, we express (3) by saying ω ± are ( n − 1)-dimensional Lipschitz graph rectifiable . In particular, if ω + ≪ ω − ≪ ω + , then at ω ± -a.e. x ∈ ∂ Ω, there is a unique ω ± -approximate tangent plane V ∈ G ( n , n − 1): ω ± ( B ( x , r )) ω ± ( B ( x , r ) \ Cone( x + V , α )) lim sup > 0 and lim sup = 0 r n − 1 r n − 1 r ↓ 0 r ↓ 0 for every cone around the ( n − 1)-plane x + V ⊥ .

Existence of Measure-Theoretic Tangents at Typical Points Theorem (Azzam-Mourgoglou-Tolsa-Volberg 2016) Let Ω ⊂ R n be a 2-sided domain equipped with harmonic measures ω ± on Ω ± . If ω + ≪ ω − ≪ ω + , then ∂ Ω = G ∪ N, where 1 ω ± ( N ) = 0 and H n − 1 G is locally finite, 2 ω ± G ≪ H n − 1 G ≪ ω ± G, 3 up to a ω ± -null set, G is contained in a countably union of graphs of Lipschitz functions f i : V i → V ⊥ i , V ∈ G ( n , n − 1) . In contemporary Geometric Measure Theory, we express (3) by saying ω ± are ( n − 1)-dimensional Lipschitz graph rectifiable . In particular, if ω + ≪ ω − ≪ ω + , then at ω ± -a.e. x ∈ ∂ Ω, there is a unique ω ± -approximate tangent plane V ∈ G ( n , n − 1): ω ± ( B ( x , r )) ω ± ( B ( x , r ) \ Cone( x + V , α )) lim sup > 0 and lim sup = 0 r n − 1 r n − 1 r ↓ 0 r ↓ 0 for every cone around the ( n − 1)-plane x + V ⊥ .

Existence of Measure-Theoretic Tangents at Typical Points Theorem (Azzam-Mourgoglou-Tolsa-Volberg 2016) Let Ω ⊂ R n be a 2-sided domain equipped with harmonic measures ω ± on Ω ± . If ω + ≪ ω − ≪ ω + , then ∂ Ω = G ∪ N, where 1 ω ± ( N ) = 0 and H n − 1 G is locally finite, 2 ω ± G ≪ H n − 1 G ≪ ω ± G, 3 up to a ω ± -null set, G is contained in a countably union of graphs of Lipschitz functions f i : V i → V ⊥ i , V ∈ G ( n , n − 1) . In contemporary Geometric Measure Theory, we express (3) by saying ω ± are ( n − 1)-dimensional Lipschitz graph rectifiable . In particular, if ω + ≪ ω − ≪ ω + , then at ω ± -a.e. x ∈ ∂ Ω, there is a unique ω ± -approximate tangent plane V ∈ G ( n , n − 1): ω ± ( B ( x , r )) ω ± ( B ( x , r ) \ Cone( x + V , α )) lim sup > 0 and lim sup = 0 r n − 1 r n − 1 r ↓ 0 r ↓ 0 for every cone around the ( n − 1)-plane x + V ⊥ .

Example: 2-Sided Domain with a Polynomial Singularity Figure: The zero set of Szulkin’s degree 3 harmonic polynomial p ( x , y , z ) = x 3 − 3 xy 2 + z 3 − 1 . 5( x 2 + y 2 ) z Ω ± = { p ± > 0 } is a 2-sided domain, ω + = ω − (pole at infinity), d ω + ≡ 0 but ∂ Ω ± = { p = 0 } is not smooth at the origin. log d ω − log d ω − d ω + is smooth �⇒ ∂ Ω is smooth

Example: 2-Sided Domain with a Polynomial Singularity Figure: The zero set of Szulkin’s degree 3 harmonic polynomial p ( x , y , z ) = x 3 − 3 xy 2 + z 3 − 1 . 5( x 2 + y 2 ) z Ω ± = { p ± > 0 } is a 2-sided domain, ω + = ω − (pole at infinity), d ω + ≡ 0 but ∂ Ω ± = { p = 0 } is not smooth at the origin. log d ω − log d ω − d ω + is smooth �⇒ ∂ Ω is smooth

Example: 2-Sided Domain with a Polynomial Singularity Figure: The zero set of Szulkin’s degree 3 harmonic polynomial p ( x , y , z ) = x 3 − 3 xy 2 + z 3 − 1 . 5( x 2 + y 2 ) z Ω ± = { p ± > 0 } is a 2-sided domain, ω + = ω − (pole at infinity), d ω + ≡ 0 but ∂ Ω ± = { p = 0 } is not smooth at the origin. log d ω − log d ω − d ω + is smooth �⇒ ∂ Ω is smooth

Useful Terminology: Local Set Approximation (B-Lewis) Let A ⊂ R n be closed, let x i ∈ A , let x i → x ∈ A , and let r i ↓ 0. If A − x → T , we say that T is a tangent set of A at x . r i Attouch-Wets topology: Σ i → Σ if and only if for every r > 0, � � lim i →∞ sup x ∈ Σ i ∩ B r dist( x , Σ) + sup y ∈ Σ ∩ B r dist( y , Σ i ) = 0 There is at least one tangent set at each x ∈ A . There could be more than one tangent set at each x ∈ A . If A − x i → S , we say that S is a pseudotangent set of A at x . r i Every tangent set of A at x is a pseudotangent set of A at x . There could be pseudotangent sets that are not tangent sets. We say that A is locally bilaterally well approximated by S if every pseudotangent set of A belongs to S .

Useful Terminology: Local Set Approximation (B-Lewis) Let A ⊂ R n be closed, let x i ∈ A , let x i → x ∈ A , and let r i ↓ 0. If A − x → T , we say that T is a tangent set of A at x . r i Attouch-Wets topology: Σ i → Σ if and only if for every r > 0, � � lim i →∞ sup x ∈ Σ i ∩ B r dist( x , Σ) + sup y ∈ Σ ∩ B r dist( y , Σ i ) = 0 There is at least one tangent set at each x ∈ A . There could be more than one tangent set at each x ∈ A . If A − x i → S , we say that S is a pseudotangent set of A at x . r i Every tangent set of A at x is a pseudotangent set of A at x . There could be pseudotangent sets that are not tangent sets. We say that A is locally bilaterally well approximated by S if every pseudotangent set of A belongs to S .

Useful Terminology: Local Set Approximation (B-Lewis) Let A ⊂ R n be closed, let x i ∈ A , let x i → x ∈ A , and let r i ↓ 0. If A − x → T , we say that T is a tangent set of A at x . r i Attouch-Wets topology: Σ i → Σ if and only if for every r > 0, � � lim i →∞ sup x ∈ Σ i ∩ B r dist( x , Σ) + sup y ∈ Σ ∩ B r dist( y , Σ i ) = 0 There is at least one tangent set at each x ∈ A . There could be more than one tangent set at each x ∈ A . If A − x i → S , we say that S is a pseudotangent set of A at x . r i Every tangent set of A at x is a pseudotangent set of A at x . There could be pseudotangent sets that are not tangent sets. We say that A is locally bilaterally well approximated by S if every pseudotangent set of A belongs to S .