Fonction propre principale optimale pour des opérateurs elliptiques avec un terme de transport grand

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1 Fonction propre principale optimale pour des opérateurs elliptiques avec un terme de transport grand Luca Rossi CAMS, CNRS - EHESS Paris Collaboration avec F. Hamel, E. Russ Luca Rossi (EHESS-CNRS) Fonction propre principale optimale 24/03/ / 10

2 Plan of the talk Plan of the talk Isoperimetric problem Faber-Krahn inequality Adding a drift Optimization in a fixed domain Optimal drift Principal eigenvalue for nonlinear operators Asymptotics of the optimal principal eigenfunction The conjectures The (partial) answers Luca Rossi (EHESS-CNRS) Fonction propre principale optimale 24/03/ / 10

3 Isoperimetric problem Isoperimetric problem for the principal eigenvalue Let λ Ω denote the principal eigenvalue of in a bounded domain Ω, i.e., ϕ = λ Ω ϕ in Ω ϕ = 0 on Ω ϕ > 0 in Ω. Question (Rayleigh 1890) Which Ω minimizes λ Ω under the constraint Ω = 1? Answer (Faber, Krahn 1920s): Ω = the ball. Tool: Schwarz symmetrization. Luca Rossi (EHESS-CNRS) Fonction propre principale optimale 24/03/ / 10

4 Isoperimetric problem Adding a drift v L (Ω). Let λ Ω,v denote the principal eigenvalue: ϕ v ϕ = λ Ω,v ϕ in Ω ϕ = 0 on Ω ϕ > 0 in Ω. Question Which Ω and v minimize λ Ω,v under the constraints Ω = 1, v τ? Answer (Hamel-Nadirashvili-Russ, Ann. of Math. 2011): Ω = the ball, Tool: new type of symmetrization. v(x) = τ x x. Luca Rossi (EHESS-CNRS) Fonction propre principale optimale 24/03/ / 10

5 Optimization in a fixed domain Optimization in a fixed domain Let Ω be a given bounded smooth domain. For v L (Ω), let λ v be the principal eigenvalue: ϕ v ϕ = λ v ϕ in Ω ϕ = 0 on Ω ϕ > 0 in Ω. τ 0, λ(τ) := inf{λ v : v L (Ω) τ}. Theorem (Hamel-Nadirashvili-Russ) The infimum is achieved by a unique v. Furthermore v = τ ϕ τ ϕ τ whenever ϕ τ 0, where ϕ τ is the associated principal eigenfunction. Luca Rossi (EHESS-CNRS) Fonction propre principale optimale 24/03/ / 10

6 Optimization in a fixed domain Principal eigenvalue for nonlinear operators ϕ τ C 2 (Ω) satisfies the nonlinear eigenvalue problem ϕ τ τ ϕ τ = λ(τ)ϕ τ in Ω ϕ τ = 0 on Ω ϕ τ > 0 in Ω. Can we say that λ(τ) is THE principal eigenvalue? Difficulty: nonlinear = Krein-Rutman theory does not directly apply. But: 1-homogeneous = Berestycki-Nirenberg-Varadhan approach does! λ gen = sup{λ : ( τ λi ) admits a positive supersolution}. Luca Rossi (EHESS-CNRS) Fonction propre principale optimale 24/03/ / 10

7 Optimization in a fixed domain Pucci (Felmer-Quaas) Bellman (Quaas-Sirakov) p and Laplacian (Kawohl-Lindqvist, Birindelli-Demengel, Juutinen) Truncated Laplacian (Birindelli-Galise-Ishii) Arbitrary degenerate ellipticity (Berestycki-Capuzzo Dolcetta- Porretta-R.) Proposition λ(τ) = max{λ : ϕ > 0, ϕ τ ϕ λϕ in Ω} = min{λ : ϕ > 0, ϕ τ ϕ λϕ in Ω, ϕ = 0 on Ω}. Furthermore, the above extrema are attained only by (a multiple of) ϕ τ. Luca Rossi (EHESS-CNRS) Fonction propre principale optimale 24/03/ / 10

8 Asymptotics of the optimal principal eigenfunction Asymptotics as τ + Theorem (Hamel-Nadirashvili-Russ) e Rτ λ(τ) e rτ, B r (x 1 ) Ω B R (x 2 ). d( ) := dist(, Ω) C := cut locus = {points where d is not differentiable} Conjecture 1 ( ) Let (x τ ) τ be maximal points for (ϕ τ ) τ. Then d(x τ ) max d Ω as τ +. Conjecture 2 ( ) x / C, ϕ τ (x) ϕ τ (x) d(x) 0 as τ +. Luca Rossi (EHESS-CNRS) Fonction propre principale optimale 24/03/ / 10

9 Asymptotics of the optimal principal eigenfunction Answers Theorem (Hamel-R.-Russ) Let (x τ ) τ be maximal points for (ϕ τ ) τ. Then d(x τ ) max d Ω as τ +. Theorem ( ) For any M > 0, sup d<m/τ ϕ τ ϕ τ d 0 as τ +. Theorem ( ) lim τ + ϕ τ (x) ϕ τ (1 e τd(x) = 1, uniformly w.r.t. x Ω. ) Luca Rossi (EHESS-CNRS) Fonction propre principale optimale 24/03/ / 10

10 Asymptotics of the optimal principal eigenfunction Idea of the proof of Theorem 1 Construction of barriers (positive supersolutions) increasing w.r.t. d: r < max d, τ 1, Ψ C 2 ((0, r)) C 0 ([0, r]) positive increasing, Ω Ψ (τ + ε)ψ λ(τ)ψ in (0, r) with ε > 0 independent of τ (using λ(τ) e rτ because B r (x 1 ) Ω). ψ := Ψ d Lemma ψ τ ψ λ(τ)ψ + (ετ d) ψ for d < r. } {{ } >0? sup d <. Ω Luca Rossi (EHESS-CNRS) Fonction propre principale optimale 24/03/ / 10