Proof by bootstrapping

Transcription

1 Proof by bootstrapping Jordan Bell Department of Mathematics, University of Toronto May 4, 2015 The Oxford English Dictionary defines to bootstrap as the following: To make use of existing resources or capabilities to raise (oneself) to a new situation or state; to modify or improve by making use of what is already present. The Picard theorem [17, p. 14, Theorem 1.17]: Theorem 1. Let M be a finite dimensional Hilbert space. Let F : M M be locally Lipschitz. Let t 0 R and let u 0 M. Then there exist T < t 0 < T + + such that, for I = (T, T + ), there exists a unique u : I M satisfying u(t 0 ) = u 0 and t u(t) = F (u(t)), t I. If T + is finite then u(t) M as t T +, and if T is finite then u(t) M as t T. Taylor s formula: Theorem 2. If f C k (B r (0)), then for all x B r (0) we have where R k (x) = k f(x) = α =k For x B r (0) we have x α α! R k (x) α =k α k 1 0 x α α! ( α f)(0) x α + R k (x), α! For k = 2 we can write Taylor s formula as: (1 t) k 1 (( α f)(tx) ( α f)(0))dt. sup ( α f)(tx) ( α f)(0). 0t1 1

2 Corollary 3. If f C 2 (B r (0)), then for all x B r (0) we have where f(x) = f(0) + Df(0)(x) D2 f(0)(x, x) + R 2 (x), R 2 (x) n2 2 x 2 sup ( α f)(tx) ( α f)(0). α =2,0t1 Thus, for any ɛ > 0 there is some r > 0 such that if x B r (0) then R 2 (x) ɛ x 2. 1 Potential well example Theorem 4. Let M be a finite dimensional Hilbert space and let V Cloc 2 (M) be such that V (0) = 0, DV (0) = 0, and D 2 V (0) is positive definite. Let N = M M. There is some δ > 0 such that if (m 1, m 2 ) N < δ then there is a unique u Cloc 1 (R, N) such that ( ) ( ) ( ) u1 u2 m1 t =, u(0) =. u 2 V (u 1 ) And u is bounded. Proof. Define F : N N by F (x, y) = ( ) y. V (x) F is locally Lipschitz, so by Picard s theorem there exist T < 0 < T + + such that, for I = (T, T + ), there exists a unique u : I N satisfying u(0) = (m 1, m 2 ) and t u(t) = F (u(t)), t I. If T + is finite then u(t) N as t T +, and if T is finite then u(t) N as t T. We shall show that u(t) N is bounded on I, which will show that T + = + and T =. Define E : I R by E(t) = 1 2 u 2(t) 2 M + V (u 1 (t)). m 2 We have de dt (t) = u 2(t), t u 2 (t) + t u 1 (t), DV (u 1 (t)) = u 2 (t), DV (u 1 (t)) + u 2 (t), DV (u 1 (t)) = 0. 2

3 This gives us the following conservation law: for all t I we have E(t) = E(0) = 1 2 m 2 2 M + V (m 1 ). Since D 2 V (0) is a symmetric positive definite matrix, there is an orthonormal basis of R n whose elements are eigenvectors for D 2 V (0) with positive eigenvalues. It follows that D 2 V (0)(v, v) λ v 2 for all v R n, where λ is the smallest eigenvalue of D 2 V (0). Let ɛ = λ 4 and let r > 0 be such that if x M < r then R 2 (x) ɛ x 2. For such x we have V (x) = V (0) + DV (0)(x) D2 V (0)(x, x) + R 2 (x) λ x 2 ɛ x 2 = 1 4 λ x 2. Let H(t) be the statement and let C(t) be the statement u(t) N r 2, u(t) N r 4. Let L = max{2, 4 r2 λ }, and let δ > 0 be small enough such that both E(0) and δ r 2. We have that H(0) is true. If H(t) is true, then u 1 (t) M r 2 < r and hence u(t) 2 N = u 1 (t) 2 M + u 2 (t) 2 M 16L 4 λ V (u 1(t)) + u 2 (t) 2 M ( L V (u 1 (t)) + 1 ) 2 u 2(t) 2 M = LE(t) = LE(0) r2 16, and hence C(t) is true. If C(t) is true, then for all t in a neighborhood of t, H(t ) is true. And if t k I converges to t I and C(t k ) is true for each k, then C(t) is true. Then by the bootstrap argument, C(t) is true for all t I. Thus, lim u(t) N r t T + 2 <, and it follows that T + = +. It likewise follows that T =. 3

4 2 Hamiltonian The following is from [17, p. 32, Exercise 1.29]. Coercive Hamiltonian implies global existence. Theorem 5. Let M be a finite dimensional symplectic vector space and let H Cloc 2 (M) be such that H(0) = 0, DH(0) = 0, and D2 H(0) is positive definite. There is some δ > 0 such that if u 0 M < δ then there is a unique u Cloc 1 (R, M) such that And u is bounded. t u = X H (u), u(0) = u 0. Proof. X H : M M is locally Lipschitz, so by Picard s theorem T < 0 < T + + such that, for I = (T, T + ), there exists a unique u : I M satisfying u(0) = u 0 and t u(t) = X H (u(t)), t I. If T + is finite then u(t) M as t T +, and if T is finite then u(t) M as t T. We shall show that u(t) M is bounded on I, which will show that T + = + and T =. Since D 2 V (0) is a symmetric positive definite matrix, it follows that D 2 V (0)(v, v) λ v 2 for all v R 2n, where λ is the smallest eigenvalue of D 2 V (0). Let ɛ = λ 4 and let r > 0 be such that if x M < r then R 2 (x) ɛ x 2. For such x we have H(x) = H(0) + DH(0)x D2 H(0)(x, x) + R 2 (x) λ x 2 ɛ x 2 = 1 4 λ x 2. Let H(t) be the statement and let C(t) be the statement u(t) N r 2, u(t) N r 4. Let δ > 0 be small enough that both H(u 0 ) λr2 64 and δ r 2. We have that H(0) is true. 4

5 If H(t) is true, then u(t) M r 2 < r and hence u(t) 2 M 4 λ H(u(t)) = 4 λ H(u 0) r2 16, and hence C(t) is true. If C(t) is true, then for all t in a neighborhood of t, H(t ) is true. And if t k I converges to t I and C(t k ) is true for each k, then C(t) is true. Then by the bootstrap argument, C(t) is true for all t I. Thus, lim u(t) M r t T + 2 <, and it follows that T + = +. It likewise follows that T =. Chipot [3, p. 227, 16.4]. Anh. 1 Grubb [10] [14, p. 231] [1]: ellliptic regularity. Rendall [16, 10.3]. proof of the stability of Minkowski space by Christodoulou and Klainerman and the theorem on formation of trapped surfaces by Christodoulou [11, p. 475] [18, p. 11, 1.7] Let φ : [0, T ] [0, ). If φ(0) α and for t such that φ(t) α we have φ(t) α/2, then φ(t) α/2 for all t [0, T ]. References [1] Michèle Audin and Mihai Damian, Morse theory and Floer homology, Universitext, Springer, London; EDP Sciences, Les Ulis, 2014, Translated from the 2010 French original by Reinie Erné. [2] Kung-Ching Chang, Methods in nonlinear analysis, Springer Monographs in Mathematics, Springer, [3] Michel Chipot, Elliptic equations: An introductory course, Springer, [4] James Colliander and Tristan Roy, Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic NLS on R 2, Commun. Pure Appl. Anal. 10 (2011), no. 2, MR (2011m:35348) 1 5

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