Quantitative homogenization: Degenerate environments and stochastic interface model Paul Dario Université Paris-Dauphine and École Normale Supérieure June 18th, 2019 Paul Dario PhD Defense June 18th, 2019 1 / 33
Introduction 1 Homogenization on percolation clusters 2 Homogenization applied to the ∇ φ model 3 Paul Dario PhD Defense June 18th, 2019 2 / 33
Contents Introduction 1 Homogenization on percolation clusters 2 Homogenization applied to the ∇ φ model 3 Paul Dario PhD Defense June 18th, 2019 3 / 33
Introduction Study of the elliptic equation ∇ ⋅ a ( x )∇ u = 0 , Paul Dario PhD Defense June 18th, 2019 4 / 33
Introduction Study of the elliptic equation ∇ ⋅ a ( x ) ∇ u = 0 , where S( R d ) , → a ∶ = { R d a ( x ) ↦ x with S( R d ) is the set of symmetric matrices; Paul Dario PhD Defense June 18th, 2019 4 / 33
Introduction Study of the elliptic equation ∇ ⋅ a ( x ) ∇ u = 0 , where S( R d ) , → a ∶ = { R d a ( x ) ↦ x with S( R d ) is the set of symmetric matrices; A uniform ellipticity assumption: there exist 0 < λ ≤ Λ < ∞ , for each x ∈ R d , λ I d ≤ a ( x ) ≤ Λ I d . Paul Dario PhD Defense June 18th, 2019 4 / 33
Introduction We assume that the environment a is random with two assumptions Stationarity: For each z ∈ Z d , the environments a ( z + ⋅ ) and a have the same law. Paul Dario PhD Defense June 18th, 2019 5 / 33
Introduction We assume that the environment a is random with two assumptions Stationarity: For each z ∈ Z d , the environments a ( z + ⋅ ) and a have the same law. Ergodicity: Ergodicity, mixing properties, concentration inequalities etc. Paul Dario PhD Defense June 18th, 2019 5 / 33
Introduction Goal: Prove that there exists a deterministic matrix a and a deterministic function u solution of ∇ ⋅ a ∇ u = 0 , such that u is close to u . Paul Dario PhD Defense June 18th, 2019 6 / 33
Introduction Goal: Prove that there exists a deterministic matrix a and a deterministic function u solution of ∇ ⋅ a ∇ u = 0 , such that u is close to u . Historical background: Qualitative theory: in the 80’s with Kozlov, Papanicolaou, Varadhan, Yurinski˘ ı etc. Paul Dario PhD Defense June 18th, 2019 6 / 33
Introduction Goal: Prove that there exists a deterministic matrix a and a deterministic function u solution of ∇ ⋅ a ∇ u = 0 , such that u is close to u . Historical background: Qualitative theory: in the 80’s with Kozlov, Papanicolaou, Varadhan, Yurinski˘ ı etc. Quantitative theory: in the 10’s with Gloria, Otto, Neukamm, Armstrong, Mourrat, Kuusi etc. Paul Dario PhD Defense June 18th, 2019 6 / 33
Three directions: 1 Homogenization on percolation clusters; 2 Homogenization of differentail forms; 3 Homogenization applied to ∇ φ model. Paul Dario PhD Defense June 18th, 2019 7 / 33
Contents Introduction 1 Homogenization on percolation clusters 2 Homogenization applied to the ∇ φ model 3 Paul Dario PhD Defense June 18th, 2019 8 / 33
Homogenization on percolation clusters Initial Goal: Extend the theory by relaxing λ I d ≤ a ( x ) ≤ Λ I d . �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� this assumption Paul Dario PhD Defense June 18th, 2019 9 / 33
Homogenization on percolation clusters Initial Goal: Extend the theory by relaxing λ I d ≤ a ( x ) ≤ Λ I d . �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� this assumption Some assumptions: Discrete setting: space Z d , discrete gradient, discrete laplacian etc. Paul Dario PhD Defense June 18th, 2019 9 / 33
Homogenization on percolation clusters Initial Goal: Extend the theory by relaxing λ I d ≤ a ( x ) ≤ Λ I d . �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� this assumption Some assumptions: Discrete setting: space Z d , discrete gradient, discrete laplacian etc. The environment a is defined on the edges ∇ ⋅ a ∇ u ( x ) = ∑ a ({ x , y })( u ( y ) − u ( x )) ; y ∼ x Paul Dario PhD Defense June 18th, 2019 9 / 33
Homogenization on percolation clusters Initial Goal: Extend the theory by relaxing λ I d ≤ a ( x ) ≤ Λ I d . �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� this assumption Some assumptions: Discrete setting: space Z d , discrete gradient, discrete laplacian etc. The environment a is defined on the edges ∇ ⋅ a ∇ u ( x ) = ∑ a ({ x , y })( u ( y ) − u ( x )) ; y ∼ x The environment a takes only two values: 0 or 1, a ∶ ↦ { 0 , 1 } ; E d ��� Set of edges Paul Dario PhD Defense June 18th, 2019 9 / 33
Homogenization on percolation clusters Initial Goal: Extend the theory by relaxing λ I d ≤ a ( x ) ≤ Λ I d . �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� this assumption Some assumptions: Discrete setting: space Z d , discrete gradient, discrete laplacian etc. The environment a is defined on the edges ∇ ⋅ a ∇ u ( x ) = ∑ a ({ x , y })( u ( y ) − u ( x )) ; y ∼ x The environment a takes only two values: 0 or 1, a ∶ ↦ { 0 , 1 } ; E d ��� Set of edges The random variables a ( e ) are i.i.d and characterized by the value p ∶ = P ( a ( e ) = 1 ) . Paul Dario PhD Defense June 18th, 2019 9 / 33
Homogenization on percolation clusters Figure 1: supercritical with p = 0 . 7 Paul Dario PhD Defense June 18th, 2019 10 / 33
Homogenization on percolation clusters We want to study harmonic function on C ∞ . ☞ Related to the behavior of the random walk on C ∞ : 1 Invariance principles Sidoravicius, Snitzman 04, Berger, Biskup 07, Mathieu, Piatnitski 07 etc. 2 Gaussian bounds on the heat kernel Mathieu, Remy 04, Barlow 04. 3 Local limit theorem Barlow, Hambly 09. Paul Dario PhD Defense June 18th, 2019 11 / 33
Homogenization on percolation clusters Theorem: Quantitative Homogenization on C ∞ There exist two exponents α > 0 and s > 0 and a nonnegative random variable X satisfying P (X > r ) ≤ exp (− cr s ) , such that ➢ for every R > 0 such that R ≥ X , Paul Dario PhD Defense June 18th, 2019 12 / 33
Homogenization on percolation clusters Theorem: Quantitative Homogenization on C ∞ There exist two exponents α > 0 and s > 0 and a nonnegative random variable X satisfying P (X > r ) ≤ exp (− cr s ) , such that ➢ for every R > 0 such that R ≥ X , ➢ for every C ∞ -harmonic function u ∶ C ∞ ∩ B R → R , Paul Dario PhD Defense June 18th, 2019 12 / 33
Homogenization on percolation clusters Theorem: Quantitative Homogenization on C ∞ There exist two exponents α > 0 and s > 0 and a nonnegative random variable X satisfying P (X > r ) ≤ exp (− cr s ) , such that ➢ for every R > 0 such that R ≥ X , ➢ for every C ∞ -harmonic function u ∶ C ∞ ∩ B R → R , ➢ there exists an harmonic function u ∶ B R → R such that ∥ u − u ∥ L 2 (C ∞ ∩ B R / 2 ) ≤ CR − α ∥ u ∥ L 2 (C ∞ ∩ B R ) . Paul Dario PhD Defense June 18th, 2019 12 / 33
Homogenization on percolation clusters We let r →∞ r − k − 1 ∥ u ∥ L 2 ( B r ) = 0 } A k ∶= { u ∶ C ∞ → R ∶ u is C ∞ -harmonic and lim and also A k ∶= { Harmonic polynomials of degree less than k } . The space A k is finite dimensional and dim A k = ( d + k − 1 ) + ( d + k − 2 ) . k − 1 k Paul Dario PhD Defense June 18th, 2019 13 / 33
Theorem (Regularity theory on C ∞ ) There exist two exponents s ,α > 0 and a nonnegative random variable X satisfying P (X > r ) ≤ exp (− cr s ) , such that the following hold: Paul Dario PhD Defense June 18th, 2019 14 / 33
Theorem (Regularity theory on C ∞ ) There exist two exponents s ,α > 0 and a nonnegative random variable X satisfying P (X > r ) ≤ exp (− cr s ) , such that the following hold: (i) For every u ∈ A k , there exists p ∈ A k such that for every r ≥ X , ∥ u − p ∥ L 2 (C ∞ ∩ B r ) ≤ Cr − α ∥ p ∥ L 2 ( B r ) ; Paul Dario PhD Defense June 18th, 2019 14 / 33
Theorem (Regularity theory on C ∞ ) There exist two exponents s ,α > 0 and a nonnegative random variable X satisfying P (X > r ) ≤ exp (− cr s ) , such that the following hold: (i) For every u ∈ A k , there exists p ∈ A k such that for every r ≥ X , ∥ u − p ∥ L 2 (C ∞ ∩ B r ) ≤ Cr − α ∥ p ∥ L 2 ( B r ) ; (ii) For every p ∈ A k , there exists u ∈ A k such that for every r ≥ X the previous estimate holds; Paul Dario PhD Defense June 18th, 2019 14 / 33
Theorem (Regularity theory on C ∞ ) There exist two exponents s ,α > 0 and a nonnegative random variable X satisfying P (X > r ) ≤ exp (− cr s ) , such that the following hold: (iii) For every C ∞ -harmonic function u , there exists φ ∈ A k such that, for every R ≥ r ≥ X , one has ∥ u − φ ∥ L 2 (C ∞ ∩ B r ) ≤ C ( r R ) k + 1 ∥ u ∥ L 2 (C ∞ ∩ B R ) . Paul Dario PhD Defense June 18th, 2019 14 / 33
Consequences and the corrector Application: Almost-surely, dim A k = dim A k = ( d + k − 1 ) + ( d + k − 2 ) . k − 1 k Paul Dario PhD Defense June 18th, 2019 15 / 33
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