# Tagged Questions

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### Indicator function of a level set

Consider a positive definite locally Lipschitz function $V:\mathbb{R}^2\to\mathbb{R}_{\geq0}$. Fix $c\in\mathbb{R}_{\geq0}$ and consider the sublevel-set $E_c=\{x:\in\mathbb{R}^2:V(x)\leq c\}$, ...
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### Can the Euler-Lagrange equations be derived from a variation over a time of order $dt$ rather than $t$?

In the calculus of variations, the solution of the Euler-Lagrange equations gives those functions for which a given functional is stationary. Now all derivations I've come across up to now, carry out ...
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I need some references about Nemytskii Mappings. Can anyone tell me some textbook about it? I am reading chapter 2 of this text www.math.tifr.res.in/~publ/ln/tifr81.pdf . And I need more results ...
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### How to prove $\int_a^b f(x)\varphi(x)dx=0\Rightarrow f(x)=0$

I am doing some reading on the calculus of variations and one of the first examples uses the following theorem: Let $f\in C[a,b]$. If $\int_a^b f(x)\varphi(x)dx=0$ for all $\varphi\in C[a,b]$, then ...
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### Does the implicit function theorem imply Peano existence theorem

In The implicit function theorem written by Krantz & Parks, it's said that the implicit function theorem implies the following existence theorem of ODE: Theorem 4.1.1 If $F(t,x)$, ...
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### Convergence of integral means of the gradient of a Sobolev function

Let $B_R(x_0)\subset\mathbb{R}^n$ with $R<1$ for $n\geq3$ and suppose $u\in H^1\big(B_R(x_0);\mathbb{R}^N\big)\cap L^{\infty}\big(B_R(x_0)\big)$. Define, \phi(R)\equiv ...
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### Calculus of variations-fields and weierstraß excess function.

if i have a lagrangian $$L (t,x(t),y(t),\dot{x}(t),\dot{y}(t))$$ that depends on two functions and one parameter. Then I will get two Euler-Lagrange equations as a test for extrema. Let us assume ...
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### Does the function need to be convex in order to satisfy the following functional?

If $\Omega \subset R^n$ and function $f(x,p)$ is strictly convex in $p$ . Is the solution to the functional $$F(u)= \int_{\Omega}f(x,Du(x))dx$$ unique in some class $C$ , and why should it be convex ...
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### Finding whether the functional attains infimum .

Given the functional $$\Phi(u) \; = \; \int_0^1 x^2.|u'(x)|^2 dx,$$ I am looking for the infimum in class $C : u\in C^1(0,1) \cap C^0[0,1]$ with end point values at $u(0)=0$ , $u(1)=1.$ The ...
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### Minimization problem of a functional.

I want to minimize the functional $$I=\int_{-1}^1u^2(x)|2x-u'(x)|^2dx$$ Here i applied and found the euler langrange equation and found the differential equation $$u'^2+2uu'-4u=4x^2$$ given is ...
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### Prove that $\int_0^1[f''(x)]^2dx\ge4.$

Let $f$ be a $C^2$ function on $[0,1]$ such that $f(0)=f(1)=f'(0)=0,f'(1)=1.$ Prove that $\int_0^1[f''(x)]^2dx\ge4.$ Find all $f$ for equality to occur.
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### Differentiating the following function.

An integral of the form : $$\int_a^b \sqrt{\frac {1+|y'(x)|^2}{2g(x-a)+v_a^2}}dx$$ The above integral is the Time taken for an object to fall under the action of gravity with some particular ...
Let $h$ be a nonnegative, smooth and convex function on $[0,1]$ and let $f(x,y):[0,1]\times[0,1]\rightarrow[0,1]$ with $f(x,y)=f(y,x)$ and $f$ continuous. Suppose I fix $r>0$ and demand that $$... 2answers 291 views ### Euler Lagrange sufficient minimization condition Suppose g \in C^1([a,b]\times\mathbb{R}\times\mathbb{R}). Let S = \{ f \in C^1([a,b]): f(a)=a_0, f(b)=b_0\}. I am trying to show that if f \in S satisfies$$ \frac{d}{dx}g_z(x,f(x),f'(x))= ...
A general problem for the Calculus of Variations asks us to minimize the value of a functional $A[f]$, where $f$ is usually a differentiable function defined on $\mathbb{R}^n$. What if, however, the ...