# Tagged Questions

Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on an infinite dimensional spaces.

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### Deduce Euler-Lagrange equation

I'm trying to get the Euler-Lagrange equation: $$\frac{d}{dx}\Big(\frac{\partial\mathfrak{L}}{\partial \dot f}\Big)-\frac{\partial \mathfrak{L}}{\partial f}=0$$ But I can't see this: Let $p$ and $q$ ...
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### Euler-Lagrange equation and unknown coefficients

I want to show that the nonlinear functional $$J(u) = \int_0^1 (u'(x))^2 + b(x)u^2(x) + f(x) u(x) \,\textrm{d}x$$ attains its minimum in exactly one point of the Sobolev space $W_0^{1,2}(0,1)$. ...
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### Characterizing Bounded Symmetric Bilinear Functions on Hilbert Spaces

Context: I am reading about Sobolev spaces and the Poisson equation from Eberhard Zeidler's Applied Functional Analysis book/article, and a key tool seems to be what Zeidler calls the "Main theorem ...
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### Geodesics in Poincare Disk

I would like to find the geodesics in the Poincare disk. I know that the metric is $$\frac{dx^2+dy^2}{(1-x^2-y^2)^2}$$ so $$s=\int \frac{\sqrt{1+y'^2}}{1-x^2-y^2}\, dx$$ Then I try to find y(x) using ...
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### Show that the function is identically Zero in certain subset

We are given a open ball D (radius = 1) in $\mathbb R^2$. and let $\{x_n\}$ be the dense sequence in the set D. Around each point $x_n$ we make a hole of radius $r_n$. The sequence $r_n$ satisfy the ...
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### Chain rule in partial derivatives

I've come across the following expression in my textbook about the chain rule in partial differentiation that I don't quite follow To be more specific, it's the diferentiation of (6.9) right at the ...
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### Ordinary differential equations as variational problems

Considering an ordinary differential equation of first order in the implicit form $$F(q(t),\dot q (t))=\alpha,\,\,\, q(0)=q_0$$ with $\alpha,\, q_0$ constants, what is the relation of the solution ...
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### For a convex function, why does lower semicontinuity imply weak lower semicontinuity?

I have come across the statement that, for a convex function, all notions of lower semicontinuity are equivalent. That is: weak lower, sequential lower, and weak sequential lower semicontinuity are ...
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### A tighter family of Markov-like inequalities

I believe there should exist tighter-than-Markov Inequalities that use the same information as the markov inequality (just expectation). Consider the proof of the markov inequality in the link below: ...
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### Calculus of variation for geodesic

I need to minimize $$J[v]=\int\sqrt{P(x)+R(x)(v')^2}dx$$ By Euler equation, I get $$\frac{d}{dx}\frac{Rv'}{\sqrt{P+Rv^{'2}}}=0$$ Then I need to solve a complex ODE, but I don't know how to deal ...
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### calculus of variations or optimize over function form

I have a question about optimizing the following quantity over function form . Given unknown function $f(\theta)$ such that $f(\theta)\geqslant 0$ and $\int f(\theta)d\theta\leq \infty$. And ...
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### Minimal surface with radially symmetrical function

The following image is from the book "Regularity Theory for Mean Curvature Flow", by Ecker. I consider the plateau problem, whose goal is to solve minimal surface given fixed boundary values. In ...
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### Calculus of Variations Problem involving mixed constraints

Motivation Let $X$ be $\mathcal{N}\Big(-\frac{\sigma^2}{2},\sigma^2\Big)$ random variable, i.e. probability density function $f(x)$ is given by f(x)=\frac{1}{ \sqrt{2\pi\sigma^2} } \...
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### Calculus of variation: Maximize fuctional

"Among the curves of given length l, at the upper half-plane, passing through the points (-a, 0) and (a, 0) , find the one that encloses the largest surface area together with the space [-a,a] ." I ...
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### How to verify this identity?

From Weinstock, "Calculus of Variations", p.24: We have the readily verifiable identity \begin{align}\frac{d}{dx}\left(y'\frac{\partial f}{\partial y'}-f\right) = y'\frac{d}{dx}\left(\frac{\...