Introduction Harnack inequalities Asymptotic behavior Boundary regularity Harnack Chains and Control Problems in Hypoelliptic Partial Differential Equations Sergio Polidoro Universit` a di Modena e Reggio Emilia Paris - September 29 - October 3, 2014 Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
Introduction Harnack inequalities Asymptotic behavior Boundary regularity Degenerate equations Consider the PDE in Ω ⊆ R N × R : m � X 2 L u ( x , t ) := k u ( x , t ) + X 0 u ( x , t ) − ∂ t u ( x , t ) = 0 , k =1 N � j ∈ C ∞ (Ω) , a k a k X k ( x ) := j ( x ) ∂ x j k = 0 , . . . , m . j =1 Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
Introduction Harnack inequalities Asymptotic behavior Boundary regularity Degenerate equations Consider the PDE in Ω ⊆ R N × R : m � X 2 L u ( x , t ) := k u ( x , t ) + X 0 u ( x , t ) − ∂ t u ( x , t ) = 0 , k =1 N � j ∈ C ∞ (Ω) , a k a k X k ( x ) := j ( x ) ∂ x j k = 0 , . . . , m . j =1 Control problem for ◮ γ ′ ( t ) = � m � � j =1 ω j ( t ) X j ( γ ( t )) + X 0 ( γ ( t )) − ∂ t Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
Introduction Harnack inequalities Asymptotic behavior Boundary regularity Focus on: ◮ Harnack inequalities ◮ [Cinti, Nystr¨ om, P.] (2010) ◮ [Cinti, Menozzi, P.] (2014) ◮ [Kogoj, Pinchover, P.] (submitted) ◮ Asymptotic bounds for positive solutions ◮ [Boscain, P.] (2007), [Cinti, P.] (2008) ◮ [Cinti, Menozzi, P.] (2014) ◮ [Garofalo, P.] (in progress) ◮ Boundary Harnack inequality for Kolmogorov equations ◮ [Cinti, Nystr¨ om, P.] (2012) ◮ [Cinti, Nystr¨ om, P.] (2013) Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
Introduction Harnack inequalities Asymptotic behavior Boundary regularity Plan of the talk ◮ Harnack inequalities for Parabolic Equations ◮ Harnack inequalities for Degenerate PDEs ◮ Asymptotic bounds for Degenerate PDEs ◮ Boundary Harnack inequality for Kolmogorov Equations Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
Introduction Harnack inequalities Asymptotic behavior Boundary regularity Euclidean setting Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
b Introduction Harnack inequalities Asymptotic behavior Boundary regularity 1 - Parabolic equations Theorem ([Pini] - 1954, [Hadamard] - 1954) Let Q r ( x , t ) = B ( x , r ) × ] t − r 2 , t [ ⊂ R n +1 , and let α, β, γ, δ ∈ ]0 , 1[ with α + β + γ < 1 . Then there exists C = C ( α, β, γ, δ, n ) such that sup u ≤ C inf u Q + r ( x , t ) Q − r ( x , t ) for every u : Q r ( x , t ) → R , u ≥ 0 , satisfying u t = ∆ u. ( x , t ) δ r α r 2 β r 2 r 2 γ r 2 δ r Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
b Introduction Harnack inequalities Asymptotic behavior Boundary regularity Equivalent formulation Theorem For any compact set K ⊂ Q 1 (0 , 0) there exists C K > 0 : sup u ≤ C K u ( x , t ) ( x , t )+ δ r K for every non-negative caloric function u : Q r ( x , t ) → R . ( x , t ) ≈ r 2 ( x , t ) + δ r K δ r ( x , t ) = ( rx , r 2 t ) Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
Introduction Harnack inequalities Asymptotic behavior Boundary regularity 2 - Bounds of the fundamental solution Theorem ([Nash - 1958] [Moser - 1964] [Aronson, Serrin - 1967]) Let Γ be the fundamental solution of N R N × R . � � � ∂ t − ∂ x i a ij ( x , t ) ∂ x j in i , j =1 Then there exist two positive constants c , C such that e − C | x − ξ | 2 e − c | x − ξ | 2 c C t − τ . ≤ Γ( x , t , ξ, τ ) ≤ t − τ N N ( t − τ ) ( t − τ ) 2 2 Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
Introduction Harnack inequalities Asymptotic behavior Boundary regularity Uniqueness in the Cauchy problem Corollary Let u be a solution to � ∂ t u ( x , t ) − � N � � ( x , t ) ∈ R N × ]0 , T [ , i , j =1 ∂ x i a ij ( x , t ) ∂ x j u ( x , t ) x ∈ R N . u ( x , 0) = 0 ◮ Upper bound ⇒ If | u ( x , t ) | ≤ Me C | x | 2 in R N × ]0 , T [ , then u ≡ 0 . ◮ Lower bound ⇒ If u ( x , t ) ≥ 0 in R N × ]0 , T [ , then u ≡ 0 . Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
Introduction Harnack inequalities Asymptotic behavior Boundary regularity 3 - Boundary behavior ◮ Divergence form Parabolic Equations ◮ [Fabes and Kenig] (1981) ◮ [Fabes and Stroock] (1986) ◮ [Garofalo] (1984) ◮ [Krylov and Safonov] (1980) ◮ [Fabes, Safonov and Yuan] (1999) ◮ Non Divergence form ◮ Fabes, Garofalo and Salsa (1986) ◮ [Fabes, Safonov (1997) ◮ [Nystr¨ om] (1997) ◮ Divergence and non Divergence ◮ [Bauman] (1984) ◮ [Caffarelli, Fabes, Mortola and Salsa] (1981) ◮ [Fabes, Garofalo, Marin-Malave and Salsa] (1988) ◮ [Jerison and Kenig] (1982) Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
b Introduction Harnack inequalities Asymptotic behavior Boundary regularity Boundary Harnack inequality Theorem ([Salsa] - 1981) Let Σ be a Lipschitz subset of the parabolic boundary of Q 1 (0 , 0) , and let K be a compact subset of Q 1 (0 , 0) such that K ∩ ∂ Q 1 (0 , 0) ⊂ Σ . Then there exists C K , Σ > 0 : sup u ≤ C K , Σ u ( x , t ) K r ( x , t ) for every non-negative solution u : Q r ( x , t ) → R to ∆ u = u t vanishing at Σ r ( x , t ) . ( x , t ) Σ r ( x , t ) K r ( x , t ) Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
Introduction Harnack inequalities Asymptotic behavior Boundary regularity Harnack inequalities for Degenerate Partial Differential Equations Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
Introduction Harnack inequalities Asymptotic behavior Boundary regularity Degenerate equations Consider the PDE in R N × R : m � X 2 L u ( x , t ) = k u ( x , t ) + X 0 u ( x , t ) − ∂ t u ( x , t ) , k =1 N � a k a k j ∈ C ∞ , X k ( x ) = j ( x ) ∂ x j k = 0 , . . . , m . j =1 Examples: � 2 + � 2 − ∂ t , � � ( x , y , s ) ∈ R 3 ◮ L := ∂ x + 2 y ∂ s ∂ y − 2 x ∂ s ◮ L := ∂ 2 ( x , y ) ∈ R 2 x + x ∂ y − ∂ t , ◮ L := ∂ 2 x + x 2 ∂ y − ∂ t , ( x , y ) ∈ R 2 Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
Introduction Harnack inequalities Asymptotic behavior Boundary regularity Regularity Theorem ([H¨ ormander] - 1967) Let u be a (distributional) solution to L u = f in Ω ⊂ R N × R . If � � = R N +1 span X 0 − ∂ t , X 1 , . . . , X m , [ X i , X j ] , . . . , [ X i , . . . , [ X j , X l ]] Then f ∈ C ∞ (Ω) u ∈ C ∞ (Ω) . ⇒ Commutators: [ X i , X j ] f := X i X j f − X j X i f Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
Introduction Harnack inequalities Asymptotic behavior Boundary regularity Example Kolmogorov operator: L := ∂ 2 x + x ∂ y − ∂ t ◮ X 1 = ∂ x , X 0 = x ∂ y , 0 1 0 X 0 − ∂ t ∼ x X 1 ∼ 0 [ X 1 , X 0 − ∂ t ] ∼ 1 − 1 0 0 Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
b Introduction Harnack inequalities Asymptotic behavior Boundary regularity Question It is possible to prove the extend the parabolic result to degenerate Kolmogorov equations? Theorem For any compact set K ⊂ Q 1 (0 , 0) there exists C K > 0 : sup u ≤ C K u ( x , t ) K r ( x , t ) for every non-negative solution u : Q r ( x , t ) → R to L u = 0 ? ( x , t ) ≈ r 2 K r ( x , t ) Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
b Introduction Harnack inequalities Asymptotic behavior Boundary regularity Strong maximum principle Theorem ([Bony] - 1969) Let u : Q → R be a non-positive solution to u xx + xu y = u t . If u (0 , 0 , 0) = 0 , then... t (0 , 0 , 0) y x
b Introduction Harnack inequalities Asymptotic behavior Boundary regularity Strong maximum principle Theorem ([Bony] - 1969) Let u : Q → R be a non-positive solution to u xx + xu y = u t . If u (0 , 0 , 0) = 0 , then... t (0 , 0 , 0) y x
b Introduction Harnack inequalities Asymptotic behavior Boundary regularity Strong maximum principle Theorem ([Bony] - 1969) Let u : Q → R be a non-positive solution to u xx + xu y = u t . If u (0 , 0 , 0) = 0 , then... t (0 , 0 , 0) y x
b b Introduction Harnack inequalities Asymptotic behavior Boundary regularity Strong maximum principle Theorem ([Bony] - 1969) Let u : Q → R be a non-positive solution to u xx + xu y = u t . If u (0 , 0 , 0) = 0 , then... t (0 , 0 , 0) (0 , 0 , 0) y y x ... u ≡ 0 in the Propagation set A (0 , 0) . Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia
Recommend
More recommend
