1. Show the following 1-d inequalities hold for any u H(I), I = [0, 1]: ||u|||(1)...

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1. Show the following 1-d inequalities hold for any u H(I), I = [0, 1]: ||u|||(1) |u||1(1) (Poincare's inequality) 1 ||U|||~ (1) == ||U||L (1) UH (1) (Sobolev's inequality) UL UL where |||| = (Su), um = (fu), u = maxzer{\u(x)]} (if u C(I)) Hint: (a) First show the inequalities hold for all u E Co (I) then use density argument. (b) For any x I, f(x) = f f' for any = Co (I) . x2 (c) For any x = 1, f, ( 1 ) ( f | ) < (1) (CUP) = x (CUP) (CUP)* < (CUF) 1. Show the following 1-d inequalities hold for any u H(I), I = [0, 1]: ||u|||(1) |u||1(1) (Poincare's inequality) 1 ||U|||~ (1) == ||U||L (1) UH (1) (Sobolev's inequality) UL UL where |||| = (Su), um = (fu), u = maxzer{\u(x)]} (if u C(I)) Hint: (a) First show the inequalities hold for all u E Co (I) then use density argument. (b) For any x I, f(x) = f f' for any = Co (I) . x2 (c) For any x = 1, f, ( 1 ) ( f | ) < (1) (CUP) = x (CUP) (CUP)* < (CUF)