变分法讲义第二章.pdfVIP

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5ƒ8ܧXC [a, b], C1 [a, b]§é ¦‚Œ ±– ±Ó$Žµ\{§ê¦8ÜS XJu\{Úꦵ 4§Ò¡S ‚ 5˜m §2 D‰‚5˜m  ÛÜ4Š §é‚5˜mÚ?ålVg 3‚5˜mþål˜„´²£ØC §§‚Ï~´ÏL‰ê½Â NS → R \ R−¡S þ‰ê§˜„P · §e 1. ∀x ∈ S, x 0 §=x ⇔ x = 0 2. ∀a ∈ R, x ∈ S, ax = |a|x 3. ∀x, y ∈ S, x + y x + y ~1. C [a, b] ) y 0 = max |y (x)| C x∈[a,b] ~2. C 1 [a, b] ) y 1 = max {|y (x)| + |y (x)|} C x∈[a,b] m m . ~3. (C [a, b]) , Y ∈ (C [a, b]) , Y = . y (x).y (x) 1 m ) Y (C [a,b])m = maxx∈[a,b] |Y (x)| = maxx∈[a,b] m y2 (x) i=1 i ~4. (C 1 [a, b])m , Y ∈ (C 1 [a, b])m ) Y (C 1 [a,b])m = maxx∈[a,b] {|Y (x)| + |Y (x)|} 1 3 ‚5˜mþ¼ 2 §3 ‚5˜mþ¼ Y ˜‚5˜m§D ⊂ Y ˜f8J : D → R §¡J ½ÂD ¼ ~1. J (y) = b sin3 x + y2 (x)dx§Y = D = C [a, b] a ~2. J (y) = b ρ(x)1 + y2 (x)dx, ρ(x) ∈ C [a, b], D = C 1 [a, b] a 1 x1 √1+y2 (x) √ √ ~3. T (y) = 2g 0 dx y (x) x1 1 1 D = y ∈ C [0, x ] y (x) 0,∀x ∈ (0, x ], dx +∞ 1