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Consider the $2\pi$-periodic inner product space $L^2[0,2\pi]$.

Let $F\triangleq\{f\in L^2[0,2\pi]|f(\theta)>0,(f(\theta),\cos\theta)=(f(\theta),\sin\theta)=0\}$.

Let $G\triangleq\{\varphi\in L^2[0,2\pi]|\varphi^\perp\cap F\neq\emptyset\}$.

Let $H\triangleq\{\varphi\in L^2[0,2\pi]|\varphi^2+(\varphi')^2<1\}$.

Then we consider the functional: $$L(\varphi)=\int_0^{2\pi}\frac{\sqrt{1-\varphi^2-(\varphi')^2}}{1-\varphi^2}d\theta$$

where $\varphi$ varies in $G\cap H$.

I have to deal with this functional in my research.

1.I can show that $L(\varphi)$ is bounded by $3\pi$ when $\varphi\in G\cap H$, and $2\pi\leq \sup L(\varphi)<3\pi$, what properties would $\varphi$ have if $L(\varphi)\geq2\pi$?

2.Note that $L(a\cos\theta+b\sin\theta)=2\pi, \forall a,b: a^2+b^2<1$, could we give any bounds on $\varphi''+\varphi$ if $L(\varphi)\geq2\pi$?

3.Can $L(\varphi)$ attains its supremum?

Could someone give me any suggestions or recommend some useful materials?

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  • $\begingroup$ I think the answer would depend on what exactly you want to do with this functional. $\endgroup$
    – Alex Degtyarev
    Dec 28, 2014 at 12:45
  • $\begingroup$ Maybe you could add some more information: where does it come from, what's your motivation, what exactly would you like to know. $\endgroup$
    – Pietro Majer
    Dec 28, 2014 at 12:48

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