Problem (F'02, #5; F'89, #6). a) Suppose that u is a continuously differentiable function on [0, 1] with u(0) = 0. Start

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Problem F 02 5 F 89 6 A Suppose That U Is A Continuously Differentiable Function On 0 1 With U 0 0 Start 1 (51.99 KiB) Viewed 52 times

Problem (F'02, #5; F'89, #6). a) Suppose that u is a continuously differentiable function on [0, 1] with u(0) = 0. Starting with u(x) = So“ u'(t) dt, prove the (sharp) estimate = = max luca)?> "\u()? dt. < (5.23) b) For any function p define p-(c) = – min{p(x),0}.21 Using the inequality (5.23), if P p is continuous on (0, 2), show that all eigenvalues of // Lu =-u" + pu on [0, 2] with u(0) = u(2) = 0 are strictly positive if sp-(t) dt < 1. =