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Homework 6

Math 442: Intro to Partial Differential Equations

(Exercises are taken from Partial Differential Equations An Introduction, Second Edition by Walter A. Strauss.)

1. Exercise §5.5 #3. Prove the inequality l R 0 l (f 0 (x))2dx ≥ [f(l) − f(0)]2 for any real function f(x) whose derivative f ’(x) is continuous. [Hint: Use Schwarz’s inequality with the pair f ’(x) and 1.]

2. Exercise §5.5 #8. Prove that both integrals in (12) tend to zero.