0 Newton Leibniz ( ) α1 ( ) αn (1) a α1,...,α n (x) u(x) = f(x) x 1 x n α 1 + +α n m 1957 Hans Lewy Lewy 1970 1 1.1 Example 1.1. (2) d 2 u dx 2 Q(x)u = f(x), u(0) = a, 1 du (0) = b. dx
Q(x), f(x) x = 0 u(x) v(x) = u(x) (a+bx) a = b = 0 Q(x) = q n x n, f(x) = f n x n, u(x) = u n x n Taylor n(n 1)u n x n 2 q j x j u k x k = f n x n. n 2 j 0 k 0 n 0 (3) (n + 2)(n + 1)u n+2 = q j u k + f n (n = 0, 1, 2,...). j+k=n a = b = 0 u 0 = u 1 = 0 (3) u 2, u 3,... u 2 (4) d 2 u dx Q(x)u = 2 u(x)2 + f(x), du u(0) = a, (0) = b. dx (4) {u n } (3) (5) (n + 2)(n + 1)u n+2 = j+k=n q j u k + j+k=n u j u k + f n (n = 0, 1, 2,...). (5) {u n } (2) (4) u(x) = u n x n (6) du j dx = f j (x, u 1, u 2,..., u n ) (j = 1,..., n), u = t (u 1,..., u n ), f = t (f 1,..., f n ) (7) du dx = f(x, u) 2
Theorem 1.2. (Cauchy) f(x, u) C n+1 Ω (x 0, a) Ω (8) u(x 0 ) = a C n (7) x 0 Example 1.1 1.2 Taylor C Theorem 1.2 C C k (9) du dx = f(x, u), u(x 0) = a R n Theorem 1.3. f(x, u) (x 0, a) R n+1 Ω f(x, u) u Lipschitz L (10) f(x, u) f(x, v) L u v (x, u), (x, v) Ω R n+1 (9) C 1 x 0 (9) (11) u(x) = a + x x 0 f(t, u(t))dt 3
Picard u 0 (x) a, (12) x u n+1 (x) = a + f(t, u n (t))dt n = 0, 1,... x 0 Theorem 1.3 f(x, u) Lipschitz f(x, u) Theorem 1.4. f(x, u) (x 0, a) R n+1 Ω (9) C 1 x 0 Theorem 1.4 Cauchy a + f(x 0, a)(x x 0 ), x [x 0, x 1 ] (13) u(x) = u(x 1 ) + f(x 1, u(x 1 ))(x x 1 ), x [x 1, x 2 ] u(x m 1 ) + f(x m 1, u(x m 1 ))(x x m 1 ), x [x m 1, x m ] x 0 < x 1 < < x m x 0 [x 0, x m ] 2 2.1 m ( ) (14) P x, u(x) := ( ) α a α (x) u(x) = f(x) x x α m α = (α 1,..., α n ), α = α 1 + + α n, ( ) α ( ) α1 ( x α = x α 1 1 xαn n, = x x 1 x n ) αn 4
(14) P (x, / x) m (14) (15) p(x, ξ) = α =m a α (x)ξ α (14) x 0 (16) p(x 0, ξ 0 ) 0 ξ 0 (16) ξ 0 (1, 0,..., 0) (16) p(x 0, (1, 0,..., 0)) 0. (14) x = x 0 x 1 x 0 (16) a (m,0,...,0) (0) 0 a (m,0,...,0) (x) x = 0 (14) (17) ( x 1 ) m u(x) + m 1 j=0 ν m j ( ) j ( ) ν a j,ν (x) u(x) = f(x). x 1 x x = (x 2,..., x n ) ν = (ν 2,..., ν n ) n 1 Cauchy Theorem 1.2 Theorem 2.1. (Cauchy-Kowalevski) (17) a j,ν (x) f(x) x = 0 x = (x 2,..., x n ) C n 1 m g k (x ) (k = 0,..., m 1) ( ) k u (18) (0, x ) = g k (x ) (k = 0,..., m 1) x k 1 (17) x = 0 (16) 5
2.2 2 u + 2 u x 2 1 x 2 2 2 u 2 u x 2 1 x 2 2 u 2 u x 1 x 2 2 = 0, = 0, = 0. x 1 x 1 > 0 x 1 < 0 20 C f(x) (14) u(x) x 0 (14) x 0 1950 Ehrenpreis Malgrange a α (x) Hörmander Acta Math., 94(1955), 161-248 Theorem 2.2. (14) x = x 0 p(x, ξ) (19) p(x 0, ξ) = 0, ξ 0 = ( ξ p)(x 0, ξ) 0 ξ = ( / ξ 1,..., / ξ n ) p(x, ξ) (14) x 0 (14) 6
(16) (16) p(x, ξ) Theorem 2.2 Hörmander p(x, ξ) (20) {Re p, Im p}(x, ξ) = 1 {p, p}(x, ξ) = 0 2i Re p, Im p p p p {f, g} f g Poisson n ( f g (21) {f, g}(x, ξ) = f ) g ξ j x j x j ξ j j=1 Hörmander (20) (14) Hans Lewy 3 3.1 Hans Lewy Hans Lewy 1957 Ann. of Math., 66(1957), 155-158 (22) u x 1 i u x 2 + 2i(x 1 + ix 2 ) u x 3 = φ (x 3 ). Theorem 3.1. (Lewy) (22) φ(x 3 ) C (22) C 1 φ(x 3 ) x 3 = 0 Theorem 3.1 (22) φ C (22) C 1 Theorem 3.1 (22) Remark 3.2. (22) C 2 Cauchy- Riemann Lewy (22) 7
Lewy (22) p(x.ξ) = ξ 1 iξ 2 + 2i(x 1 + ix 2 )ξ 3 {Re p, Im p} = 4ξ 3 0 Hörmander Lewy Hörmander (20) 3.2 Nirenberg-Treves Lewy Hörmander (23) p(x, ξ) = 0 = {Re p, Im p}(x, ξ) 0 Mizohata (24) u + ix j u 1 = f(x) x 1 x 2 j 0 j 1970 Nirenberg Treves Comm. Pure Appl. Math., 23(1970), 1-38, 459-509 Definition 3.3. (14) p(x, ξ) x = x 0 Re(ap) Im(ap) + p(x, ξ) x 0 (Ψ) Im(ap) (P ) a 0 a = 1 a = i Re p Re p Hamilton (25) dx j dt = (Re p) ξ j, dξ j dt = (Re p) x j (Re p)(x(t), ξ(t)) = 0 Mizohata (24) a = 1 Re p (x 1 (t), x 2 (t); ξ 1 (t), ξ 2 (t)) = (t + x 0 1, x0 2 ; 0, ξ0 2 ) x0 1, x0 2, ξ0 2 Re p Im p t Im p = ξ2 0(t + x0 1 )j j Poisson {Re p, Im p} Re p Hamilton Im p Hörmander (23) Re p Im p Im p Nirenberg Treves (Ψ) (P ) 8
Nirenberg-Treves (P ) (P ) 1960 (Ψ) p(x, ξ) = ( 1) m p(x, ξ) (P ) (Ψ) Nirenberg-Treves 1973 Beals-Fefferman (P ) (Ψ) 1978 Moyer 1981 Hörmander (Ψ) 1988 Lerner (Ψ) Dencker 2006 Nirenberg-Treves Dencker 2005 Clay (cf. http://www.claymath.org/research award/dencker/) Nirenberg-Treves 4 4.1 R n C n Ω C f(x) ( ) α (26) a α (x) u(x) = f(x) x α m C u(x) Ω Example 4.1. R 2 Ω 9
x 2 Ω 0 x 1 (27) u x 1 = 1 x 2 1 +. x2 2 1/(x 2 1 + x 2 2) Ω C Ω C (27) (27) (28) u(x 1, x 2 ) = 1 ( ) Tan 1 x1 + v(x 2 ) v(x 2 ) x 2 x 2 x 2 Ω C u(x 1, x 2 ) ε (29) u(ε, x 2 ) u( ε, x 2 ) x 2 x 2 = 0 C (28) v(x 2 ) (29) x 2 = 0 C (27) Ω C (27) Ω (27) x 1 x 1 (27) Ω (27) x 1 Ω x 1 Ω 10
4.2 (27) x 1 (30) a 1 (x) u + + a n (x) u = f(x) a j (x) x 1 x n (31) dx 1 dt = a 1(x),..., dx n dt = a n (x) (x 1 (t),..., x n (t)) (30) (x 1 (t),..., x n (t)) (32) d dt [u(x 1(t),..., x n (t))] = f(x 1 (t),..., x n (t)) (30) (31) (27) x 1 (31) (30) (26) 1970 (26) Fact 4.2. (26) u(x) Hamilton Fact 4.2 u(x) (x(t), ξ(t)) (x, ξ) x(t) x (26) u(x) x x 4.3 4.1 Example 4.1 x 1 (26) 4.1 (26) Ω 11
1. C a α (x) Ω p (26) Ω Duistermaat- Hörmander; Acta Math., 128(1972), 183-269 2. Ω p 3. Ω p Ω (26) Ω Suzuki; Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A, 11(1972), 253-258 Kawai-Takei; Adv. in Math., 80(1990), 110-133 3.2 [5], [6] Fourier C [4] References [1] 6 1994 [2] 1977 [3] 1965 [4] L. Hörmander: The Analysis of Linear Partial Differential Operators, Volume I-IV, Springer-Verlag, 1983-1985 Volume IV, Section 26. [5] M. Sato, T. Kawai and M. Kashiwara: Microfunctions and pseudo-differential equations, Lecture Notes in Mathematics, No. 287, Springer, 1973, pp. 265-529. [6] 18 1980 12