Strichartz inequalities for waves in a strictly convex domain Oana - PowerPoint PPT Presentation

Strichartz inequalities for waves in a strictly convex domain Oana Ivanovici ( † ), Richard Lascar ( ‡ ), Gilles Lebeau ( † ) and Fabrice Planchon ( † ) ( † ) Universit´ e Nice Sophia Antipolis ( ‡ ) Universit´ e Paris 7 lebeau@unice.fr In honor of Johannes SJOSTRAND 25 September, 2013 Gilles Lebeau (Universit´ e Nice Sophia Antipolis) Strichartz 25 September, 2013 1 / 26

Outline Result 1 The parametrix construction 2 Dispersive estimates 3 Interpolation estimates 4 Optimality of the result 5 Comments 6 Gilles Lebeau (Universit´ e Nice Sophia Antipolis) Strichartz 25 September, 2013 2 / 26

Outline Result 1 The parametrix construction 2 Dispersive estimates 3 Interpolation estimates 4 Optimality of the result 5 Comments 6 Gilles Lebeau (Universit´ e Nice Sophia Antipolis) Strichartz 25 September, 2013 3 / 26

Strichartz in R d Dispersion � χ ( hD t ) e ± it √ x ≤ Ch − d min (1 , ( h |△| ( δ a ) � L ∞ t ) α d ) (1.1) Strichartz ( ∂ 2 t − △ ) u = 0 h β � χ ( hD t ) u � L q x ) ≤ C ( � u (0 , x ) � L 2 + � hD t u (0 , x ) � L 2 ) (1.2) t ∈ [0 , T ] ( L r q ∈ ]2 , ∞ [ , r ∈ [2 , ∞ ] 1 q = α d (1 2 − 1 β = ( d − α d )(1 2 − 1 r ) , r ) with α d = d − 1 in the free space R d 2 Gilles Lebeau (Universit´ e Nice Sophia Antipolis) Strichartz 25 September, 2013 4 / 26

Main result Let ( M , g ) be a Riemannian manifold. Let Ω be an open relatively compact subset of M with smooth boundary ∂ Ω. We assume that Ω is Strictly Convex in ( M , g ), i.e any (small) piece of geodesic tangent to ∂ Ω is exactly tangent at order 2 and lies outside Ω. We denote by △ the Laplacian associated to the metric g on M . Theorem For solutions of the mixed problem ( ∂ 2 t − △ ) u = 0 on R t × Ω and u = 0 on R t × ∂ Ω , the Strichartz inequalities hold true with α d = d − 1 − 1 6 , d = dim ( M ) 2 Remark This was proved by M. Blair, H.Smith and C.Sogge in the case d = 2 for arbitrary boundary (i.e without convexity assumption). The above theorem improves all the known results for d ≥ 3 . Gilles Lebeau (Universit´ e Nice Sophia Antipolis) Strichartz 25 September, 2013 5 / 26

Outline Result 1 The parametrix construction 2 Dispersive estimates 3 Interpolation estimates 4 Optimality of the result 5 Comments 6 Gilles Lebeau (Universit´ e Nice Sophia Antipolis) Strichartz 25 September, 2013 6 / 26

The problem is local near any point p 0 of the boundary. In geodesic coordinates normal to ∂ Ω and after conjugation by a non vanishing smooth function e ( x , y ), one has for ( x , y ) ∈ R × R d − 1 near (0 , 0) △ = e − 1 · △ · e = ∂ 2 ˜ x + R ( x , y , ∂ y ) Ω = { x > 0 } , p 0 = ( x = 0 , y = 0) 0n the boundary, in geodesic coordinates centered at y = 0, one has � ∂ 2 y j + O ( y 2 ) R 0 ( y , ∂ y ) = R (0 , y , ∂ y ) = Let R 1 ( y , ∂ y ) = ∂ x R (0 , y , ∂ y ) = � R j , k 1 ( y ) ∂ y j ∂ y k . The quadratic form � R j , k 1 ( y ) η j η k is positively define. We introduce the Model Laplacian � � � R j , k △ M = ∂ 2 � ∂ 2 x + y j + x 1 (0) ∂ y j ∂ y k Gilles Lebeau (Universit´ e Nice Sophia Antipolis) Strichartz 25 September, 2013 7 / 26

Set ρ ( ω, η ) = ( η 2 + ω q ( η ) 2 / 3 ) 1 / 2 , � R j , k q ( η ) = 1 (0) η j η k The following theorem is due to Melrose-Taylor, Eskin, Zworski, ... Theorem There exists two phases ψ ( x , y , η, ω ) homogeneous of degree 1 , ζ ( x , y , η, ω ) homogeneous of degree 2 / 3 , and symbols p 0 , 1 ( x , y , η, ω ) of degree 0 ( ω is 2 / 3 homogeneous, and | ω | η | − 2 / 3 | is small) such that G ( x , y ; η, ω ) = e i ψ � � p 0 Ai ( ζ ) + xp 1 | η | − 1 / 3 Ai ′ ( ζ ) satisfy − ˜ △ G = ρ 2 G + O C ∞ ( | η | −∞ ) near ( x , y ) = (0 , 0) ζ = − ω + x | η | 2 / 3 e 0 ( x , y , η, ω ) with p 0 and e 0 elliptic near any point (0 , 0 , η, 0) with η ∈ R d − 1 \ 0 . Gilles Lebeau (Universit´ e Nice Sophia Antipolis) Strichartz 25 September, 2013 8 / 26

Let ( X − x ) u + ( Y − y ) v + Γ( X , Y , u , v ) be a generating function for a (Melrose) canonical transformation χ M such that χ M ( x = 0 , ξ 2 + η 2 + xq ( η ) = 1) = ( X = 0 , Ξ 2 + R ( X , Y , Θ) = 1) near Σ 0 = { ( x , y , ξ, η ) , x = 0 , y = 0 , ξ = 0 , | η | = 1 } . One has Γ(0 , Y , u , v ) is independent of u There exists a symbol q ( x , y , η, ω, σ ) of degree 0 ( σ is 1 / 3 homogeneous) compactly supported near N 0 = { x = 0 , y = 0 , ω = 0 , σ = 0 , η ∈ R d − 1 \ 0 } and elliptic on N 0 such that e i ( y η + σ 3 / 3+ σ ( xq ( η ) 1 / 3 − ω )+ ρ Γ( x , y , σ q ( η )1 / 3 G ( x , y ; η, ω ) = 1 � , η ρ )) q d σ ρ 2 π Gilles Lebeau (Universit´ e Nice Sophia Antipolis) Strichartz 25 September, 2013 9 / 26

Let G ( t , x , y ; a ) be the Green function solution of the mixed problem, with a ∈ ]0 , a 0 ], a 0 > 0 small ( ∂ 2 t − ˜ △ ) G = 0 in x > 0 , G| x =0 = 0 G| t =0 = δ x = a , y =0 , ∂ t G| t =0 = 0 Definition Let χ ( x , t , y , hD t , hD y ) be a h-pseudo differential (tangential) operator of degree 0 , compactly supported near Σ 0 = { x = 0 , t = 0 , y = 0 , τ = 1 , | η | = 1 } and equal to identity near ˜ ˜ Σ 0 . A ”parametrix” is an approximation (near { x = 0 , y = 0 , t = 0 } ) mod 0 C ∞ ( h ∞ ) , and uniformly in a ∈ ]0 , a 0 ] of χ ( x , t , y , hD t , hD y )( G ( . ; a )) . Gilles Lebeau (Universit´ e Nice Sophia Antipolis) Strichartz 25 September, 2013 10 / 26

Set ω = h − 2 / 3 α . Recall ρ ( α, θ ) = ( θ 2 + α q ( θ ) 2 / 3 ) 1 / 2 . Let Φ( x , y , θ, α, s ) be the phase function Φ = y θ + s 3 / 3 + s ( xq ( θ ) 1 / 3 − α ) + ρ ( α, θ )Γ( x , y , sq ( θ ) 1 / 3 θ ρ ( α, θ ) , ρ ( α, θ )) and let q h ( x , y , θ, α, s ) = h − 1 / 3 q ( x , y , h − 1 θ, h − 2 / 3 α, h − 1 / 3 s ) Then J ( f )( x , y ) = 1 � i h (Φ − y ′ θ − t ′ α ) q h f ( y ′ , t ′ ) dy ′ dt ′ d θ d α ds e 2 π is a semiclassical OIF associated to a canonical transformation χ such that χ ( { y ′ = 0 , t ′ = 0 , | η ′ | = 1 , τ ′ = 0 } ) = { y = 0 , x = 0 , | η | = 1 , ξ = 0 } Moreover, J is elliptic on the above set and − h 2 ˜ △ J ( f ) = J ( ρ 2 ( hD t ′ , hD y ′ ) f ) mod O C ∞ ( h ∞ ) Gilles Lebeau (Universit´ e Nice Sophia Antipolis) Strichartz 25 September, 2013 11 / 26

Airy-Poisson summation formula Let A ± ( z ) = e ∓ i π/ 3 Ai ( e ∓ i π/ 3 z ). One has Ai ( − z ) = A + ( z ) + A − ( z ). For ω ∈ R , set L ( ω ) = π + i log( A − ( ω ) A + ( ω )) The function L is analytic, strictly increasing, L (0) = π/ 3, lim ω →−∞ L ( ω ) = 0, L ( ω ) ≃ 4 ω 3 / 2 ( ω → + ∞ ), and one has ∀ k ∈ N ∗ 3 � ∞ L ′ ( ω k ) = Ai 2 ( x − ω k ) dx L ( ω k ) = 2 π k ⇔ Ai ( − ω k ) = 0 , 0 Lemma The following equality holds true in D ′ ( R ω ) . 1 e − iNL ( ω ) = 2 π � � L ′ ( ω k ) δ ω = ω k N ∈ Z k ∈ N ∗ Gilles Lebeau (Universit´ e Nice Sophia Antipolis) Strichartz 25 September, 2013 12 / 26

Let g h , a ( y ′ , t ′ ) such that J ( g h , a ) − 1 2 δ x = a , y =0 = R with WF h ( R ) ∩ W = ∅ , where W is a fixed neighborhood of { ( x = 0 , y = 0 , ξ = 0 , η ) , | η | = 1 } . For ω ∈ R , set (recall α = h 2 / 3 ω ) K ω ( f )( t , x , y ) = h 2 / 3 � i h ( t ρ ( h 2 / 3 ω,θ )+Φ − y ′ θ − t ′ h 2 / 3 ω ) q h f ( y ′ , t ′ ) dy ′ dt ′ d θ ds e 2 π � One has J ( f ) = R K ω ( f ) | t =0 d ω . Finally, set 1 � e − iNL ( ω ) , K ω ( g h , a ) > D ′ ( R ) = P h , a ( t , x , y ) = 2 π � < L ′ ( ω k ) K ω k ( g h , a ) N ∈ Z k ∈ N ∗ Proposition P h , a ( t , x , y ) is a parametrix. The proof uses the left formula for a ≥ h 2 / 3 − ǫ , and the right formula for a ≤ h 4 / 7+ ǫ . Gilles Lebeau (Universit´ e Nice Sophia Antipolis) Strichartz 25 September, 2013 13 / 26

Outline Result 1 The parametrix construction 2 Dispersive estimates 3 Interpolation estimates 4 Optimality of the result 5 Comments 6 Gilles Lebeau (Universit´ e Nice Sophia Antipolis) Strichartz 25 September, 2013 14 / 26

△ = △ M , with q ( η ) = | η | 2 (Friedlander model), where The special case ˜ one has of course Γ = 0, has been studied by Ivanovici-Lebeau-Planchon in Dispersion for waves inside strictly convex domains I: the Friedlander model case. (http://arxiv.org/abs/1208.0925 and to appear in Annals of Maths). The analysis of phase integrals are (essentially) the same in the general case, and leads to the following result. Theorem 1 , ( h � d − 2 � |P h , a ( t , x , y ) | ≤ Ch − d min 2 C t ) (3.1) C = ( h t ) 1 / 2 + a 1 / 8 h 1 / 4 for a ≥ h 2 / 3 − ǫ C = ( h t ) 1 / 3 for a ≤ h 1 / 3+ ǫ Corollary Strichartz holds true in any dimension d ≥ 2 with α d = d − 1 − 1 2 4 Gilles Lebeau (Universit´ e Nice Sophia Antipolis) Strichartz 25 September, 2013 15 / 26