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I would like to find the greatest lower bound and hopefully the function that minimizes $\int_0^1 [f’’(x)]^2 dx$ where $f\in C^2([0,1])$ with $f(0) = f(1) = 0$ and $f’(0) = 1$.

The only thing I could think of to bound the integral was Cauchy-Schwarz: $$\left(\int_0^1 f’’(x)dx\right)^2 \leq \int_0^1 (1)^2dx\int_0^1 [f’’(x)]^2dx \\ \implies \int_0^1[f’’(x)]^2dx \geq [f’(1) – f’(0)]^2 = [f’(1) – 1]^2$$

But this involves $f’(1)$ which is not known.

AlRacoon
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  • Perhaps extending $f$ to a periodic function with period $1$ and using Fourier series might provide an answer. – Kavi Rama Murthy Jan 07 '20 at 23:24
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    This is a near-duplicate of the following questions: https://math.stackexchange.com/questions/219139/ https://math.stackexchange.com/questions/1257404/ https://math.stackexchange.com/questions/1264629/ – Maximilian Janisch Jan 07 '20 at 23:28

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