0
votes
1answer
37 views

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 $E_c=\{x:\in\mathbb{R}^2:V(x)\leq c\}$, ...
1
vote
0answers
31 views

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 ...
0
votes
1answer
17 views

References about Nemytskii Mappings

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 ...
0
votes
1answer
62 views

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 ...
3
votes
0answers
57 views

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 $F(t,x)$, ...
1
vote
1answer
48 views

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, \begin{equation} \phi(R)\equiv ...
10
votes
4answers
305 views

When is the moment of inertia of a smooth plane curve is maximum?

Given a smooth plane curve $(x(s),y(s))$, parameterized in arc length $s$, of fixed finite length $L$, its moment of inertia about its center of mass (axis perpendicular to the plane) is given as $$MI ...
2
votes
1answer
90 views

Euler Lagrange equation derivation and application of the fundamental lemma of the calculus of variations

Say we have: (1) $J(x) = \int_{\textit{to}}^{\textit{tf}} g(x(t),\dot{x}(t),t) dt$. We go through the general derivation and arrive at: (2) $\delta J(x,\delta x) = ...
0
votes
1answer
42 views

Prove that $a(u-u_{h},u-u_{h})\ge 0$

Assume that $a$ is bilinear, symmetric and positive definite form, $u\in X$ and $u_{h}\in X_{h}\subset X$. I know the following fact: $$a(u-u_{h},u_{h})=0$$ Frm positive definiteness ...
0
votes
1answer
35 views

Finding extremal of function J

Find a curve passing through $\left(0,0\right)$ and $\left(1,1\right)$ that is an extremal for the functional $\displaystyle{{\rm J}\left(x,y,y'\right) = \int\left\{\left[y'(x)\right]^{2} + ...
2
votes
1answer
107 views

Finding critical points of functional (Euler equations)

Consider for $T>0$ the functional $$u\mapsto J(u) := \int_0^T (\dot{u}(t)^2-u(t)^2)dt. $$ on the space $W_0^{1,\infty}((0,T),\mathbb{R})$. (a) Depending on $T$, find the critical points of $J$ ...
4
votes
1answer
217 views

Existence of a Lagrange multiplier (Euler Lagrange equations + holonomic constraints )

Let $I=[a,b]\subset \mathbb{R}, G:\mathbb{R}^n\to \mathbb{R}^k$ smooth, $0<k<n, M=G^{-1}(0)$. Assume that $DG(x)$ has full rank for all $x\in M$. Fix $p_1,p_2\in M$ and assume $u\in ...
3
votes
1answer
111 views

Dido's problem with Euler equations

I'm considering Dido's problem: Consider 2 differentiable arcs $C$ and $C_0$ in $\mathbb{R}^2$ from the point $P$ to $Q$ and back. We keep $C_0,P,Q$ fixed, and want to choose the arc $C$ such that ...
2
votes
1answer
66 views

Inverse problem in calculus of variations

I am interested in knowing which differential equations follow from a variational principle. I am reading this and it provides the answer for ordinary differential equations. Is there a complete ...
1
vote
2answers
52 views

Existence of solution in Hölder spaces

Let's say we have a PDE, for example the Laplace equation: $$ \Delta u = f. $$ Usually, to solve such a thing, one finds its variational formulation, and solves it in some Sobolev space. My question ...
5
votes
1answer
56 views

Show that $ M$ is constant on $[a,b]$ (variational calculus)

Let $F:\mathbb{R}^n\times \mathbb{R}^n \to \mathbb{R}$ be $C^2$ on $[a,b]$ and $u$ be a solution for the Euler-lagrange equations for the functional given by $$J(u) = \int F(u(t),\dot{u}(t)).dt, $$ ...
1
vote
1answer
57 views

Piecewise $C^1$ function is element of $W^{1,\infty}$

Hey I'm confused about the following (apparantly) fact: Let $u:[a,b]\to\mathbb{R}$ a piecewise $C^1$ function, i.e. there exists $a=t_1<t_2<\cdots < t_n = b$ such that $u|_{[t_i,t_{i+1}]} ...
3
votes
2answers
100 views

Derive the solution to the Lagrangian $ \mathcal L= y(x)\sqrt{1+y'(x)^2}$

I am supposed to derive the solution to the Lagrangian $$ \mathcal L= y(x)\sqrt{1+y'(x)^2}$$ Unfortunately I am unable to solve both, the Euler Lagrange equation or the Beltrami equation. It may be ...
0
votes
1answer
84 views

Gateaux derivative

I have the following definition of Gateaux differentiability $f$ is Gateaux differentiable at $x_0$ if there is a continuous and linear operator $T$ so that $$ \lim_{t \rightarrow ...
0
votes
0answers
66 views

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 ...
0
votes
1answer
23 views

Identity between functions

Let $f$ and $g$ be continuous functions of one real variable. We want to show that $\frac{\mathrm{d}}{\mathrm{d}t}f = g$ on the interval $[a,b]$. I have shown that for any subinterval $[t_a,t_b] ...
2
votes
1answer
58 views

Show that the path of shortest hyperbolic length satisfies $(x-c)^2+y^2=r^2$

The hyperbolic length of a curve $y:[a,b]\rightarrow\mathbb{RxR}_+$ is given by the functional $$\lambda(y)=\int_a^b\frac{\sqrt{1+y'^2}}{y}dx$$ Show that the path of shortest hyperbolic length ...
1
vote
2answers
93 views

Show that the infimum of a functional is zero, but this infimum is never achieved.

Show that the infimum of the integrals $$\int_0^1(y'^2(x)-1)^2dx$$ among all $y(x)\in C^2[0,1]$ such that $y(0)=y(1)=0$, is zero, but is not achieved by any function in this set. What I've worked on: ...
0
votes
1answer
168 views

Weak Minimizer of a Functional

I showed that $u(x) = \frac{x^2}{2}$ is a potential minimizer for the functional $\int_0^2 \frac{n}{2}u'(x)^2-nu(x) \, dx$ in $C^2[0,2]$ with $u(0) = 0$ and $u(2)=2$ where $n$ is a positive constant ...
3
votes
0answers
67 views

Different functional brachystochrone

Until today I thought that $$ \int_0^b \sqrt{\frac{1+y'(x)^2}{2gy(x)}} dx$$ would be the only functional to derive the brachystochrone, but in the textbook Variational Methods in Mathematical Physics ...
3
votes
1answer
127 views

Computing the Euler Lagrange equations

Let $F(u) = \int_0^1(u'')^2+u^2dx $ on $C^2[0,1]$ satisfying $u(0)=a,u(1)=b,u'(0)=c,u'(1)=d$ where $a,b,c,d \in \mathbb{R}$. If $u_*$ is a minimizer, for $\phi \in C^2[0,1],\ \frac{d}{ds}| _{s=0} ...
0
votes
0answers
29 views

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 ...
1
vote
1answer
60 views

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 ...
0
votes
1answer
90 views

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 ...
6
votes
1answer
313 views

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.
0
votes
0answers
39 views

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 ...
1
vote
0answers
82 views

Extremizing an Integral under a cyclic condition

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 $$ ...
0
votes
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))= ...
8
votes
1answer
993 views

Calculus of Variations and Lagrange Multipliers

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 ...