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Results tagged with calculus-of-variations
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user 104330
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
2
votes
Accepted
Poincaré inequality under weighted average condition
For $u\in W^{1, 2}(\Omega)$ set $L(u) = \int_\Omega e^{-ay}u(x, y)dxdy$. We want to prove that if $L(u) = 0$ then $||u||_{L^2}\le C||\nabla u||_{L^2}$ for some universal constant $C$. We will first de …
- 6,476
4
votes
Accepted
If $f\in C([0,\infty))$, does $\delta>0$ and $g\in C^1((0,\delta))\cap C([0,\delta])$ s.t. $...
Sure you can find such a $g$ and without much difficulty. Without loss of generality we assume that $f(0) = 0$ (otherwise just subtract it). Construct the sequence $\delta_n>0$ inductively in such a w …
- 6,476