The Dirichlet problem for second order parabolic operators in - PowerPoint PPT Presentation

The Dirichlet problem for second order parabolic operators in divergence form Kaj Nyström Department of Mathematics, Uppsala University Kaj Nyström , Department of Mathematics, Uppsala University The Dirichlet problem for parabolic equations

Second order parabolic equations H u = ( ∂ t + L ) u := ∂ t u − div λ, x A ( x , t ) ∇ λ, x u = 0 (0.1) = { ( λ, x , t ) = ( λ, x 1 , .., x n , t ) ∈ R n + 1 × R : λ > 0 } . in R n + 2 + n � κ | ξ | 2 ≤ A i , j ( x , t ) ξ i ξ j , | A ( x , t ) ξ · ζ | ≤ C | ξ || ζ | . (0.2) i , j = 0 A is real but no assumptions on symmetry of A . Kaj Nyström , Department of Mathematics, Uppsala University The Dirichlet problem for parabolic equations

Parabolic measure Given f ∈ C 0 ( R n + 1 ) there exists a unique weak solution to the continuous Dirichlet problem H u = 0 in R n + 2 , + u ∈ C ([ 0 , ∞ ) × R n + 1 ) , u ( 0 , x , t ) = f ( x , t ) on R n + 1 . �� u ( λ, x , t ) = R n + 1 f ( y , s ) d ω ( λ, x , t , y , s ) . ω ( λ, x , t , · ) : parabolic measure (at ( λ, x , t ) ). Kaj Nyström , Department of Mathematics, Uppsala University The Dirichlet problem for parabolic equations

Doubling property of parabolic measure Q = Q r ( x ) := B ( x , r ) ⊂ R n , I = I r ( t ) := ( t − r 2 , t + r 2 ) , ∆ = ∆ r ( x , t ) = Q r ( x ) × I r ( t ) , ℓ (∆) := r , c ∆ := cQ × c 2 I . Given r > 0 and ( x 0 , t 0 ) ∈ R n + 1 , A + r ( x 0 , t 0 ) := ( 4 r , x 0 , t 0 + 16 r 2 ) . Theorem Assume that A is real and satisfies (0.2) . If ( x 0 , t 0 ) ∈ R n + 1 , 0 < r 0 < ∞ , ∆ 0 := ∆ r 0 ( x 0 , t 0 ) , then � � � � A + A + ω 4 r 0 ( x 0 , t 0 ) , 2 ∆ � ω 4 r 0 ( x 0 , t 0 ) , ∆ whenever ∆ ⊂ 2 ∆ 0 . Kaj Nyström , Department of Mathematics, Uppsala University The Dirichlet problem for parabolic equations

Doubling property of parabolic measure Q = Q r ( x ) := B ( x , r ) ⊂ R n , I = I r ( t ) := ( t − r 2 , t + r 2 ) , ∆ = ∆ r ( x , t ) = Q r ( x ) × I r ( t ) , ℓ (∆) := r , c ∆ := cQ × c 2 I . Given r > 0 and ( x 0 , t 0 ) ∈ R n + 1 , A + r ( x 0 , t 0 ) := ( 4 r , x 0 , t 0 + 16 r 2 ) . Theorem Assume that A is real and satisfies (0.2) . If ( x 0 , t 0 ) ∈ R n + 1 , 0 < r 0 < ∞ , ∆ 0 := ∆ r 0 ( x 0 , t 0 ) , then � � � � A + A + ω 4 r 0 ( x 0 , t 0 ) , 2 ∆ � ω 4 r 0 ( x 0 , t 0 ) , ∆ whenever ∆ ⊂ 2 ∆ 0 . Kaj Nyström , Department of Mathematics, Uppsala University The Dirichlet problem for parabolic equations

Definition: A ∞ -property of parabolic measure Definition Let ( x 0 , t 0 ) ∈ R n + 1 , 0 < r 0 < ∞ , ∆ 0 := ∆ r 0 ( x 0 , t 0 ) . We say � � A + ω 4 r 0 ( x 0 , t 0 ) , · ∈ A ∞ (∆ 0 , d x d t ) if ∀ ε > 0 ∃ δ = δ ( ε ) > 0 such that if E ⊂ ∆ for some ∆ ⊂ ∆ 0 , then � � A + ω 4 r 0 ( x 0 , t 0 ) , E | E | � � < δ = ⇒ | ∆ | < ε. A + ω 4 r 0 ( x 0 , t 0 ) , ∆ � � A + ω ∈ A ∞ ( d x d t ) if ω 4 r 0 ( x 0 , t 0 ) , · ∈ A ∞ (∆ 0 , d x d t ) for all ∆ 0 as above and with uniform constants. If ω ∈ A ∞ ( d x d t ) , then � � � � A + A + d ω 4 r 0 ( x 0 , t 0 ) , x , t = K 4 r 0 ( x 0 , t 0 ) , x , t d x d t . Kaj Nyström , Department of Mathematics, Uppsala University The Dirichlet problem for parabolic equations

Main Result Theorem Assume that A is real and satisfies (0.2) . Then parabolic mea- sure ω belongs to A ∞ ( d x d t ) with constants depending only n and the ellipticity constants. The result holds under no assumptions on A = A ( x , t ) besides (0.2): t -dependent coefficients are allowed! The result is new even in the case when A ( x , t ) is symmetric. L 2 results hold under stronger structural assumptions. Kaj Nyström , Department of Mathematics, Uppsala University The Dirichlet problem for parabolic equations

Main Result Theorem Assume that A is real and satisfies (0.2) . Then parabolic mea- sure ω belongs to A ∞ ( d x d t ) with constants depending only n and the ellipticity constants. The result holds under no assumptions on A = A ( x , t ) besides (0.2): t -dependent coefficients are allowed! The result is new even in the case when A ( x , t ) is symmetric. L 2 results hold under stronger structural assumptions. Kaj Nyström , Department of Mathematics, Uppsala University The Dirichlet problem for parabolic equations

Proof of the main result - references S. Hofmann, C. Kenig, S. Mayboroda and J. Pipher. Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators. J. Amer. Math. Soc. 28 (2015), 483–529. C. Kenig, B. Kirchheim, J. Pipher and T. Toro. Square functions and absolute continuity of elliptic measure. J. Geom. Anal. 26 (2016), no. 3, 2383–2410. AEN. L 2 well-posedness of boundary value problems and the Kato square root problem for parabolic systems with measurable coefficients. To appear in J. Eur. Math. Soc. AEN. The Dirichlet problem for second order parabolic operators in divergence form. To appear in Journal de l’Ecole polytechnique - Mathematiques. Kaj Nyström , Department of Mathematics, Uppsala University The Dirichlet problem for parabolic equations

Proof of the main result - references S. Hofmann, C. Kenig, S. Mayboroda and J. Pipher. Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators. J. Amer. Math. Soc. 28 (2015), 483–529. C. Kenig, B. Kirchheim, J. Pipher and T. Toro. Square functions and absolute continuity of elliptic measure. J. Geom. Anal. 26 (2016), no. 3, 2383–2410. AEN. L 2 well-posedness of boundary value problems and the Kato square root problem for parabolic systems with measurable coefficients. To appear in J. Eur. Math. Soc. AEN. The Dirichlet problem for second order parabolic operators in divergence form. To appear in Journal de l’Ecole polytechnique - Mathematiques. Kaj Nyström , Department of Mathematics, Uppsala University The Dirichlet problem for parabolic equations

Main result: reduction to a Carleson measure estimate To conclude that ω ∈ A ∞ ( d x d t ) it suffices to prove Theorem � � for some Borel set S ⊂ R n + 1 . Then Let u ( λ, x , t ) = ω λ, x , t , S u satisfies the following Carleson measure estimate for all parabolic cubes ∆ ⊂ R n + 1 : � ℓ (∆) �� |∇ λ, x u | 2 λ d x d t d λ � | ∆ | . (0.3) 0 ∆ ‘Proof’: Given E ⊂ ∆ , δ > 0, there exists K ( δ ) , such that if � � � � A + A + ω 4 r 0 ( x 0 , t 0 ) , E < δω 4 r 0 ( x 0 , t 0 ) , ∆ , then there exists a set � � S , E ⊂ S ⊆ ∆ , such that if u ( λ, x , t ) := ω λ, x , t , S , then � ℓ (∆) �� |∇ λ, x u | 2 λ d x d t d λ. K ( δ ) | E | � 0 ∆ Kaj Nyström , Department of Mathematics, Uppsala University The Dirichlet problem for parabolic equations

Main result: reduction to a Carleson measure estimate To conclude that ω ∈ A ∞ ( d x d t ) it suffices to prove Theorem � � for some Borel set S ⊂ R n + 1 . Then Let u ( λ, x , t ) = ω λ, x , t , S u satisfies the following Carleson measure estimate for all parabolic cubes ∆ ⊂ R n + 1 : � ℓ (∆) �� |∇ λ, x u | 2 λ d x d t d λ � | ∆ | . (0.3) 0 ∆ ‘Proof’: Given E ⊂ ∆ , δ > 0, there exists K ( δ ) , such that if � � � � A + A + ω 4 r 0 ( x 0 , t 0 ) , E < δω 4 r 0 ( x 0 , t 0 ) , ∆ , then there exists a set � � S , E ⊂ S ⊆ ∆ , such that if u ( λ, x , t ) := ω λ, x , t , S , then � ℓ (∆) �� |∇ λ, x u | 2 λ d x d t d λ. K ( δ ) | E | � 0 ∆ Kaj Nyström , Department of Mathematics, Uppsala University The Dirichlet problem for parabolic equations

Main result: reduction to a Carleson measure estimate It suffices to prove that � ℓ (∆) �� | ∂ λ u | 2 λ d x d t d λ � | ∆ | , (0.4) ∆ 0 for all parabolic cubes ∆ . To prove (0.4) it is enough to prove that the following holds: for each parabolic cube ∆ ⊂ R n + 1 , r := ℓ (∆) , there is a Borel set F ⊂ 16 ∆ with | ∆ | � | F | , such that � r �� | ∂ λ u | 2 λ d x d t d λ � | ∆ | . (0.5) 0 F Reduction: need to construct F and verify (0.5). Kaj Nyström , Department of Mathematics, Uppsala University The Dirichlet problem for parabolic equations

Main result: reduction to a Carleson measure estimate It suffices to prove that � ℓ (∆) �� | ∂ λ u | 2 λ d x d t d λ � | ∆ | , (0.4) ∆ 0 for all parabolic cubes ∆ . To prove (0.4) it is enough to prove that the following holds: for each parabolic cube ∆ ⊂ R n + 1 , r := ℓ (∆) , there is a Borel set F ⊂ 16 ∆ with | ∆ | � | F | , such that � r �� | ∂ λ u | 2 λ d x d t d λ � | ∆ | . (0.5) 0 F Reduction: need to construct F and verify (0.5). Kaj Nyström , Department of Mathematics, Uppsala University The Dirichlet problem for parabolic equations

Main result: reduction to a Carleson measure estimate It suffices to prove that � ℓ (∆) �� | ∂ λ u | 2 λ d x d t d λ � | ∆ | , (0.4) ∆ 0 for all parabolic cubes ∆ . To prove (0.4) it is enough to prove that the following holds: for each parabolic cube ∆ ⊂ R n + 1 , r := ℓ (∆) , there is a Borel set F ⊂ 16 ∆ with | ∆ | � | F | , such that � r �� | ∂ λ u | 2 λ d x d t d λ � | ∆ | . (0.5) 0 F Reduction: need to construct F and verify (0.5). Kaj Nyström , Department of Mathematics, Uppsala University The Dirichlet problem for parabolic equations