Some problems involving fractional order operators

Transcription

1 Some problems involving fractional order operators Universitat Politècnica de Catalunya December 9th, 2009

2 Definition ( ) γ Infinitesimal generator of a Levy process Pseudo-differential operator, principal symbol ξ 2γ. ( ) γ f (x) = C n,γ R n f (x) f (ξ) dξ x ξ n+2γ Caffarelli-Silvestre s extension: For γ (0, 1), γ = 1 a 2, { div(y a u) = 0 in R n+1 + u = f, at y = 0

3 Definition ( ) γ Infinitesimal generator of a Levy process Pseudo-differential operator, principal symbol ξ 2γ. ( ) γ f (x) = C n,γ R n f (x) f (ξ) dξ x ξ n+2γ Caffarelli-Silvestre s extension: For γ (0, 1), γ = 1 a 2, { div(y a u) = 0 in R n+1 + u = f, at y = 0 ( ) γ f = d n,γ lim y 0 y a y u

4 Definition ( ) γ Infinitesimal generator of a Levy process Pseudo-differential operator, principal symbol ξ 2γ. ( ) γ f (x) = C n,γ R n f (x) f (ξ) dξ x ξ n+2γ Caffarelli-Silvestre s extension: For γ (0, 1), γ = 1 a 2, { div(y a u) = 0 in R n+1 + u = f, at y = 0 ( ) γ f = d n,γ lim y 0 y a y u Dirichlet-to-Neumann operator, non-local γ = 1/2 (half-laplacian)

5 Conformally covariant operators Riemannian manifold: (M n, g) Conformal angle preserving Conformal change of metric: g = fg, f > 0

6 Conformally covariant operators Riemannian manifold: (M n, g) Conformal angle preserving Conformal change of metric: g = fg, f > 0 Def: A g is a conformally covariant operator of bidegree (a, b) if under the change of metric g w = e 2w g, A gw φ = e bw A g (e aw φ) for all φ C (M).

7 Conformally covariant operators Riemannian manifold: (M n, g) Conformal angle preserving Conformal change of metric: g = fg, f > 0 Def: A g is a conformally covariant operator of bidegree (a, b) if under the change of metric g w = e 2w g, A gw φ = e bw A g (e aw φ) for all φ C (M). Associated curvature: Q A??

8 Conformally covariant operators Riemannian manifold: (M n, g) Conformal angle preserving Conformal change of metric: g = fg, f > 0 Def: A g is a conformally covariant operator of bidegree (a, b) if under the change of metric g w = e 2w g, A gw φ = e bw A g (e aw φ) for all φ C (M). Associated curvature: Q A?? Examples: Conformal Laplacian (order 2), L g := g + c n R g Paneitz operator (order 4), P g := ( g ) Order 2k (Graham-Jenne-Mason-Sparling)

9 Conformally compact Einstein manifolds (X n+1, g) smooth manifold of dimension n + 1 with metric g Smooth boundary M n = X Defining function: ρ > 0 in X, ρ = 0 on M, dρ 0 on M.

10 Conformally compact Einstein manifolds (X n+1, g) smooth manifold of dimension n + 1 with metric g Smooth boundary M n = X Defining function: ρ > 0 in X, ρ = 0 on M, dρ 0 on M. Def: (X, g) is conformally compact if there exists ρ such that (X, ḡ) is compact for ḡ = ρ 2 g. Def: Conformal infinity: (M, [h]), h = ḡ M Def: (X, g) is conformally compact Einstein if, in addition, Ric g = ng

11 Conformally compact Einstein manifolds (X n+1, g) smooth manifold of dimension n + 1 with metric g Smooth boundary M n = X Defining function: ρ > 0 in X, ρ = 0 on M, dρ 0 on M. Def: (X, g) is conformally compact if there exists ρ such that (X, ḡ) is compact for ḡ = ρ 2 g. Def: Conformal infinity: (M, [h]), h = ḡ M Def: (X, g) is conformally compact Einstein if, in addition, Ric g = ng Model example: ( Hyperbolic space ( H n+1 ) ( ) ) 2, g H = B n+1, 2 dy 2 = 1 y 2 ( ) R n+1 +, dx 2 +dy 2 y 2

12 Scattering theory Theorem (Graham-Zworski, Mazzeo) There exists a meromorphic family of pseudo-differential operators on M, called {S(s)}, s C, Re(s) > n/2 such that σ (S(s)) = c n,s σ ( ( M ) s n/2)

13 Scattering theory Theorem (Graham-Zworski, Mazzeo) There exists a meromorphic family of pseudo-differential operators on M, called {S(s)}, s C, Re(s) > n/2 such that σ (S(s)) = c n,s σ ( ( M ) s n/2) Construction: Given f on M, solve the eigenvalue equation Then S(s)f = G M. g u s s(n s)u s = 0 in X (1) u s = Fρ s + Gρ n s, F ρ=0 = f. (2)

14 Scattering theory Theorem (Graham-Zworski, Mazzeo) There exists a meromorphic family of pseudo-differential operators on M, called {S(s)}, s C, Re(s) > n/2 such that σ (S(s)) = c n,s σ ( ( M ) s n/2) Construction: Given f on M, solve the eigenvalue equation g u s s(n s)u s = 0 in X (1) u s = Fρ s + Gρ n s, F ρ=0 = f. (2) Then S(s)f = G M. Operators of fractional order: Define P γ := c n,γ S ( n 2 + γ) Conformal: g u = u 4 n 2γ g Pγ gu = u n+2γ n 2γ Pγ g (u )

15 Scattering theory Theorem (Graham-Zworski, Mazzeo) There exists a meromorphic family of pseudo-differential operators on M, called {S(s)}, s C, Re(s) > n/2 such that σ (S(s)) = c n,s σ ( ( M ) s n/2) Construction: Given f on M, solve the eigenvalue equation g u s s(n s)u s = 0 in X (1) u s = Fρ s + Gρ n s, F ρ=0 = f. (2) Then S(s)f = G M. Operators of fractional order: Define P γ := c n,γ S ( n 2 + γ) Conformal: g u = u 4 n 2γ g Pγ gu Model example: in R n, γ Z: = u n+2γ n 2γ P g γ (u ) P γ = ( ) γ

16 Results (Chang-G.) Theorem A Caffarelli-Silvestre extension problem in R n+1 +, γ (0, 1) scattering theory in H n+1

17 Results (Chang-G.) Theorem A Caffarelli-Silvestre extension problem in R n+1 +, γ (0, 1) scattering theory in H n+1 Theorem B For γ (0, 1), on general c.c. Einstein manifolds, { div(ρ a U) + E(ρ)U = 0 in (X, ḡ) U = f, at ρ = 0 P γ f = c γ lim ρ a ρ U, γ = 1 a ρ 0 2 ( ) Here E(ρ) = ḡ ρ a 2 ρ a 2 + ( γ 2 1 4) ρ 2+a + n 1 4n R ḡρ a

18 Theorem C For γ = m + α, m N, α (0, 1) { div(y a u) = 0 in R n+1 + u = f, at y = 0 ( P γ f = c γ lim y 1 2α y y 0 ( ))) y 1 y (...y 1 y u } {{ } m+1 derivatives

19 Q γ curvature Def: Qγ g := Pγ g (1). Conformal invariance: g w = w 4 n 2γ g Pγ g (w) = w n+2γ n 2γ Qγ gw. γ Yamabe problem: find g w = w 4 n 2γ g such that Qγ gw = cst Proposition (Qing-G.) If the γ-yamabe constant of M is strictly less than S n, then there exists a minimizer for I γ [U] = X ρa U 2 + X ρa E(ρ)U 2 ( M U n 2γ 2n ) n 2γ n Second order degenerate elliptic equations Weighted Sobolev trace inequalities γ = 1/2 : constant mean curvature on the boundary (Escobar)

20 Theorem: (G.-Mazzeo-Sire) The model case S n \S k with the metric given by S n k 1 H k+1 has constant Q γ -curvature. Moreover, if k < n/2, Q γ is positive when k < n 2γ 2.

21 Renormalized volume We have that vol g + ({ρ > ɛ}) = c 0 ɛ n + c 2 ɛ n c n 1 ɛ 1 + V + o(1) Computation (Fefferman-Graham) Let u s be the solution of g u s s(n s)u s = 0 in X (3) u s = F ρ s + Gρ n s, F ρ=0 = 1. (4) Define v = d ds s=n u s. It is a solution of g +v = n in X. Integrate on ρ > ɛ. Then V = 1 Q n d n 2 M 2

22 Weighted renormalized volume How about if we use u s instead?? Then we define vol g +,γ({ρ > ɛ}) := {ρ>ɛ} (ρ ) n 2 γ dvol g + Chang-G It has an asymptotic expansion If γ < 1 2, then [ vol g +,γ({ρ > ɛ}) = ɛ n 2 γ ( ] n 2 + γ) 1 vol(m) + ɛ 2γ V γ + l.o.t. where V γ := 1 If γ > 1 2, then 1 d γ n 2 γ M Q γ. [ vol g +,γ({ρ > ɛ}) = ɛ n 2 γ ( ] n 2 + γ) 1 vol(m) + ɛw0 + l.o.t. for W 0 := ( n 2 + γ) 1 (2n 1) M J 0.

23 Thank you!!