Elliptic measure and rectifiability Tatiana Toro University of Washington Workshop on Real Harmonic Analysis and its Applications to Partial Differential Equations and Geometric Measure Theory: on the occasion of the 60th birthday of Steve Hofmann Madrid, Espa˜ na Junio 1, 2018 Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 1 / 24
Motivation What is the relationship between the geometry of a domain and the boundary regularity of the solutions to a differential operator on this domain? ( regularity=degree of smoothness .) Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 2 / 24
Motivation What is the relationship between the geometry of a domain and the boundary regularity of the solutions to a differential operator on this domain? ( regularity=degree of smoothness .) Can the regularity at the boundary of a “general harmonic function” distinguish between a rectifiable and a purely unrectifiable boundary? Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 2 / 24
Some history F&M Riesz (1916): Let Ω ⊂ R 2 be a simply connected domain bounded by a Jordan curve. If H 1 ( ∂ Ω) < ∞ then the harmonic measure ω and the surface measure σ = H ∂ Ω are mutually absolutely continuous, i.e. ω ( E ) = 0 iff σ ( E ) = 0 Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 3 / 24
Some history F&M Riesz (1916): Let Ω ⊂ R 2 be a simply connected domain bounded by a Jordan curve. If H 1 ( ∂ Ω) < ∞ then the harmonic measure ω and the surface measure σ = H ∂ Ω are mutually absolutely continuous, i.e. ω ( E ) = 0 iff σ ( E ) = 0 Lavrentiev (1936): Let Ω ⊂ R 2 be a bounded simply connected chord arc domain. Then ω ∈ A ∞ ( σ ). What happens in higher dimensions? Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 3 / 24
Domains A domain Ω ⊂ R n is uniform (1-sided NTA) (with constant M ) [Aikawa - Hofmann & Martell] if it satisfies: Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 4 / 24
Domains A domain Ω ⊂ R n is uniform (1-sided NTA) (with constant M ) [Aikawa - Hofmann & Martell] if it satisfies: ◮ Interior corkscrew condition (with constant M ) Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 4 / 24
Domains A domain Ω ⊂ R n is uniform (1-sided NTA) (with constant M ) [Aikawa - Hofmann & Martell] if it satisfies: ◮ Interior corkscrew condition (with constant M ) ◮ Harnack chain condition (with constant M ) Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 4 / 24
Domains A domain Ω ⊂ R n is uniform (1-sided NTA) (with constant M ) [Aikawa - Hofmann & Martell] if it satisfies: ◮ Interior corkscrew condition (with constant M ) ◮ Harnack chain condition (with constant M ) A domain Ω ⊂ R n is NTA (non-tangentially accessible) [Jerison - Kenig] if: ◮ Ω is uniform ◮ Exterior corkscrew condition Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 4 / 24
Domains A domain Ω ⊂ R n is uniform (1-sided NTA) (with constant M ) [Aikawa - Hofmann & Martell] if it satisfies: ◮ Interior corkscrew condition (with constant M ) ◮ Harnack chain condition (with constant M ) A domain Ω ⊂ R n is NTA (non-tangentially accessible) [Jerison - Kenig] if: ◮ Ω is uniform ◮ Exterior corkscrew condition A domain Ω ⊂ R n has Ahlfors regular boundary if there exists c 0 > 1 such that for q ∈ ∂ Ω and r ∈ (0 , diam Ω) 0 r n − 1 ≤ σ ( B ( q , r )) ≤ c 0 r n − 1 . c − 1 Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 4 / 24
Harmonic measure and quantitative rectifiability Let Ω ⊂ R n be a uniform domain with Ahlfors regular boundary. Then the following are equivalent: 1) ∂ Ω is ( n − 1)-uniformly rectifiable. Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 5 / 24
Harmonic measure and quantitative rectifiability Let Ω ⊂ R n be a uniform domain with Ahlfors regular boundary. Then the following are equivalent: 1) ∂ Ω is ( n − 1)-uniformly rectifiable. 2) Ω is an NTA domain, thus a chord arc domain. Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 5 / 24
Harmonic measure and quantitative rectifiability Let Ω ⊂ R n be a uniform domain with Ahlfors regular boundary. Then the following are equivalent: 1) ∂ Ω is ( n − 1)-uniformly rectifiable. 2) Ω is an NTA domain, thus a chord arc domain. 3) ω ∈ A ∞ ( σ ). Proof: 2) = ⇒ 3) David–Jerison & Semmes Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 5 / 24
Harmonic measure and quantitative rectifiability Let Ω ⊂ R n be a uniform domain with Ahlfors regular boundary. Then the following are equivalent: 1) ∂ Ω is ( n − 1)-uniformly rectifiable. 2) Ω is an NTA domain, thus a chord arc domain. 3) ω ∈ A ∞ ( σ ). Proof: 2) = ⇒ 3) David–Jerison & Semmes ⇒ 1) Hofmann–Martell–Uriarte-Tuero 3) = Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 5 / 24
Harmonic measure and quantitative rectifiability Let Ω ⊂ R n be a uniform domain with Ahlfors regular boundary. Then the following are equivalent: 1) ∂ Ω is ( n − 1)-uniformly rectifiable. 2) Ω is an NTA domain, thus a chord arc domain. 3) ω ∈ A ∞ ( σ ). Proof: 2) = ⇒ 3) David–Jerison & Semmes ⇒ 1) Hofmann–Martell–Uriarte-Tuero 3) = 1) = ⇒ 2) Azzam–Hofmann–Martell–Nystr¨ om–Toro Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 5 / 24
Divergence form elliptic operators Let Ω ⊂ R n be a bounded Wiener regular domain and Lu = − div ( A ( x ) ∇ u ) with A ( x ) = ( a ij ( x )) an uniformly elliptic symmetric matrix with bounded measurable coefficients, i.e. λ | ξ | 2 ≤ � A ( x ) ξ, ξ � , � A ( x ) ξ, ζ � Λ ≤ | ξ || ζ | for x ∈ Ω and ξ, ζ ∈ R n . Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 6 / 24
Divergence form elliptic operators Let Ω ⊂ R n be a bounded Wiener regular domain and Lu = − div ( A ( x ) ∇ u ) with A ( x ) = ( a ij ( x )) an uniformly elliptic symmetric matrix with bounded measurable coefficients, i.e. λ | ξ | 2 ≤ � A ( x ) ξ, ξ � , � A ( x ) ξ, ζ � Λ ≤ | ξ || ζ | for x ∈ Ω and ξ, ζ ∈ R n . Let ω L be the corresponding elliptic measure. Recall that if f ∈ C ( ∂ Ω) there exists u ∈ C (Ω) such that � Lu = 0 in Ω (1) u = f on ∂ Ω Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 6 / 24
Divergence form elliptic operators Let Ω ⊂ R n be a bounded Wiener regular domain and Lu = − div ( A ( x ) ∇ u ) with A ( x ) = ( a ij ( x )) an uniformly elliptic symmetric matrix with bounded measurable coefficients, i.e. λ | ξ | 2 ≤ � A ( x ) ξ, ξ � , � A ( x ) ξ, ζ � Λ ≤ | ξ || ζ | for x ∈ Ω and ξ, ζ ∈ R n . Let ω L be the corresponding elliptic measure. Recall that if f ∈ C ( ∂ Ω) there exists u ∈ C (Ω) such that � Lu = 0 in Ω (1) u = f on ∂ Ω Moreover ˆ f ( q ) d ω x u ( x ) = L ( q ) ∂ Ω Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 6 / 24
Questions For what type of domains Ω and operators L do we have ω L ∈ A ∞ ( σ )? Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 7 / 24
Questions For what type of domains Ω and operators L do we have ω L ∈ A ∞ ( σ )? What does the fact that ω L ∈ A ∞ ( σ ) imply about the geometry of Ω? Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 7 / 24
Questions For what type of domains Ω and operators L do we have ω L ∈ A ∞ ( σ )? What does the fact that ω L ∈ A ∞ ( σ ) imply about the geometry of Ω? Caffarelli-Fabes-Kenig, Modica-Mortola, Modica-Mortola-Salsa (1981-2): There exist Lipschitz domains and operators L for which ω L and σ are mutually singular. Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 7 / 24
Questions For what type of domains Ω and operators L do we have ω L ∈ A ∞ ( σ )? What does the fact that ω L ∈ A ∞ ( σ ) imply about the geometry of Ω? Caffarelli-Fabes-Kenig, Modica-Mortola, Modica-Mortola-Salsa (1981-2): There exist Lipschitz domains and operators L for which ω L and σ are mutually singular. Questions: Characterize the operators L for which ω L ∈ A ∞ ( σ ) . To what extent does this characterization depend on the domain? Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 7 / 24
Different approaches Perturbation theory. Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 8 / 24
Different approaches Perturbation theory. Structure of the matrix A . Tatiana Toro (University of Washington) Elliptic measure and rectifiability June 1, 2018 8 / 24
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