Timeline for Prove/disprove $(\int_{0}^{2 \pi} \!\!\cos f(x) d x)^{2}+(\int_{0}^{2 \pi}\!\!\! \sqrt{(f'(x))^{2}+\sin ^{2} f(x)}dx)^{2}\ge 4\pi^{2}$
Current License: CC BY-SA 4.0
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| Jan 29, 2021 at 14:47 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Added some tags and a comment about partially defined solutions.
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| Jan 29, 2021 at 14:43 | comment | added | Robert Bryant | @mlk: Indeed, there are issues that I haven't treated or discussed. Linear growth in $f'$ by itself is not really an issue: For example, the minimizers of $\int_a^b\sqrt{f'(x)^2+1}\,dx$ are all smooth, and the integrand there has linear growth. More serious is when the integrand is not strictly convex in $f'$, as it is not in this case when $\sin f(x)$ vanishes. (This is also where the equation (1) is singular, which is not a coincidence.) However, such local regularity issues arise frequently in geometrically meaningful minimization problems, and there are methods for treating them. | |
| Jan 29, 2021 at 12:37 | comment | added | mlk | Regarding the calculus of variations, these arguments indeed exist, but because the integrand of $b$ only has linear growth in $f'$, your minimizer might end up in $BV$, i.e. have jumps. | |
| Jan 29, 2021 at 12:00 | history | answered | Robert Bryant | CC BY-SA 4.0 |