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Results tagged with calculus-of-variations
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user 108824
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
2
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0
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212
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Is $f$ defined by $f(x) = t\mapsto G(t , x(t))$ differentiable?
Let us consider $X = AC([0 , 1] , \mathbb{R}^n)$, and $Y=L^{1} ([0,1] , \mathbb{R}^n )$ as Banach spaces with their usual norms. Let $G: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ be a continuo …
- 369
1
vote
1
answer
996
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Converting an integral equation into a differential equation
Let $a, b \in \mathbb R^n$ and $f, g \in L^1 [0,1]$. Assume for all $h \in AC[0,1 ]$ (space of absolutely continuous functions) following integral equality holds
$$ \int_{0}^{1} \langle f(t) , h(t) …
- 369
2
votes
1
answer
213
views
Is $ f(x(.)) := \int_{0}^{1} F ( x(t)) \; dt$ differentiable?
Let $f : AC[0, 1] \to R$ be defined by $ f(x(.)) := \int_{0}^{1} F ( x(t)) \; dt$. Where, $AC[0, 1]$ is the set of absolutely continuous functions with the norm $W^{1,1}$, and $F: R^n \to R$ is conti …
- 369