2
votes
1answer
29 views

Continuity of a functional with respect to two different norms

Let $J$ be a functional defined on $E = C^1[a,b]$ by $$J(y) = \int_a^b \sqrt{1 + (y^{\prime}(x))^2} \, dx.$$ Define the following two norms on $E$: $$\|y\|_{\infty} = \max_{a\leq x\leq b} |y(x)|$$ ...
1
vote
0answers
46 views

Application of a general “Weierstrass theorem”

http://books.google.at/books?id=9OSrV73a40gC&pg=PA45&lpg=PA45 gives a general Weierstrass theorem. Are there notable applications of this theorem, say in the calculus of variations? (I could ...
0
votes
1answer
17 views

Lebesgue integral question using du Boise-Reymond lemma

This question was inspired a previous question of mine. If we are given that $\Omega \subset \mathbb{R}^{n}$ is open and bounded and $$\int_{\Omega}fv dx = 0$$ where $f \in C(\Omega)$ and $v \in ...
0
votes
1answer
61 views

Minimizing cost function (Eikonal)

Given a cost function $F(x_{1},x_{2},x_{3})$ and a starting Point $S \in \mathbb{R}^{3}$ we define a function $T$ as $T(x,y,z)=\min_{\gamma} \int_{0}^{1} F(\gamma(t))dt$ such that $\gamma(0)=S$ and ...
2
votes
2answers
52 views

Equivalent norm in Sobolev space

Let $\rho\in H^{1}(0,\pi)$ be a function, and consider the functional $$ I(\rho)=\bigg(\int_{0}^{\pi}{\sqrt{\rho^2(t)+\dot\rho^2(t)}\,dt}\bigg)^2. $$ I'm asking if it is equivalent to the norm $$ ...
1
vote
0answers
26 views

Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
0
votes
2answers
25 views

Quadratic Minimization

Consider a functional $I\colon H \to R$ on $H$ Banach space, sufficiently regular. Is in generally true that $$ \inf_{\rho \in H}{I^2(\rho)}=\Big(\inf_{\rho \in H}{I(\rho)} \Big)^2 \quad ? $$ If ...
1
vote
1answer
44 views

Optimal String Shape Problem

So here is the problem I am working on, Given a curve of length L connecting the points (0,1) and (1,0) find an expression for the equation of the curve that minimizes the area underneath it. In ...
1
vote
0answers
12 views

Minimization Problem and Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
0
votes
1answer
60 views

Eikonal Equation [duplicate]

Given a cost function $F(x_{1},x_{2},x_{3})$ and a starting Point $A \in \mathbb{R}^{3}$ we define a function $T$ as $T(x,y,z)=\min_{\gamma} \int_{0}^{1} F(\gamma(t))dt$ such that $\gamma(0)=A$ and ...
1
vote
0answers
25 views

On a Variational Inequality

Let $H$ be a Hilbert space with real inner product. Consider $f: C \rightarrow C$, where $C \subset H$ is closed and convex. I am not sure about the variational inequality problem: find $x \in H$ ...
0
votes
1answer
25 views

Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
0
votes
0answers
10 views

How does invariance of $q$ wrt $\lambda$ for a stationary functional, restrict the function?

Suppose I have the following functional: $$S(q) = \int_{b}^{a}L(t, q(t), q'(t))dt$$ and $q(t) = x(t) + \lambda$, where $\lambda$ is a constant independent of t. If $S(q)$ is stationary for a ...
1
vote
0answers
38 views

Existence of a Minimizer $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $

given the following functional $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $ with $\rho>0$ , $\|\rho\|_1 = 1$ and obviously $\rho\in L^1(\mathbb R^3)$. Can I see ...
0
votes
1answer
22 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 ...
2
votes
0answers
37 views

Minimizing a functional by variation

I have a problem at the last step of my proof. I have the following functional to be minimized on $\rho\in L^1(\mathbb R^d)$. Here $\lambda$ is a Lagrange multiplier and $\rho\geq 0$. $h(\rho) = ...
2
votes
1answer
84 views

Extremum of functional of a complex function

consider functional $E$ defined by $$E[z]=\int F(x,z(x))dx$$ where $F$ is a complex-valued nonlinear function. How can we find the function $z(x)$ so that $$G=|E|^2=EE^*=\iint ...
1
vote
0answers
60 views

Test functions on a compact interval

Consider a functional $E:C([0,1]) \rightarrow \mathbb{R}$ of the form $$E(g) = \int_0^1 g(s)ds$$ In dealing with such functionals one often needs test functions. If one talks about the space ...
3
votes
0answers
71 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 $F(t,x)$, ...
0
votes
1answer
42 views

Total variation for functions, Meaning of supremum as used here?

On Wikipedia article, here: http://en.wikipedia.org/wiki/Total_variation, on definition 1.1 there says, "where the supremum runs over the set of all partitions ..." AFAIK supremum is defined for a ...
1
vote
0answers
58 views

Calculus of variations: the inside function has an integral

It is known that if the functional $$J=\int_a^b L(x,f(x))dx \tag{1}$$ has an extremum, then the Euler equation $\frac{\partial{L}}{\partial{f(x)}}=0$ holds. My question is, for example, what if ...
2
votes
1answer
113 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) = ...
1
vote
1answer
134 views

Maximum Entropy (The existence of a Calculus of Variations problem)

Take maximum differential entropy as an example: Gaussian achieves the maximum differential entropy when the second order moment is fixed. The calculus of variation form: \begin{equation} ...
1
vote
0answers
31 views

Derivative of infimum in variational problem

Let $\mathcal{E}(\phi,\alpha), \phi\in \mathcal{D}$ be a functional on some domain $\mathcal{D}$ that depends on a parameter $\alpha$. In the expression $$\frac{\partial}{\partial \alpha} \inf_{\phi ...
0
votes
0answers
23 views

Exchange limit and infimum in variational problem

Let $\{\mathcal{E}_n\}_{n\in \mathbb{N}}$ be a sequence of functionals over the same domain $\mathcal{D}$. What are sufficient conditions on the sequence and possibly the domain such that ...
0
votes
1answer
48 views

Prove differentiability of functional.

In $C[0;1]$ space let's consider following functional: $$\phi(f) = \int_{0}^{1}(1+f(t))^{3}dt.$$ Prove differentiability of $\phi$ and find $\mathrm{D}\phi(f)$ for: $f(t)=0$, $f(t)=t$, $f(t)=\cos t$. ...
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
0answers
92 views

Calculus of variations: Lagrange multipliers with functions depend on only one variable

The problem: \begin{align} \min & \iint k(x_1,x_2,y_1,y_2) \, dx_1 \, dx_2 \tag{1}\\ \mathrm{s.t. } & \iint h(x_1,x_2,y_1,y_2) \, dx_1 \, dx_2=l \tag{2}\\ & g(x_1,x_2,y_1,y_2)=0 ...
0
votes
1answer
73 views

Is this functional differentiable?

A functional $\Phi$ is differentiable if there exist $F$ and $R$ such that $\Phi(f+h)-\Phi(f)=F(f,h)+R(f,h)$, where $F$ depends linearly on $h$ and $R(f,h) = O(h^2)$. Define a functional $\Phi(f) = ...
6
votes
0answers
91 views

References on the Nash-Moser Implicit Function Theorem

To learn, the Nash-Moser implicit function theorem, I tried with Hamilton (1982) The Inverse Function Theorem of Nash and Moser. But, the article is very encyclopedic. I have a background in ...
2
votes
2answers
109 views

Natural growth conditions and weak solutions for inhomogenous systems.

Let $\Omega\subset \mathbb{R}^n, \ n\geq 2$ be a bounded Lipschitz domain. Consider the following inhomogenous system (in divergence form) subject to zero Dirichlet boundary conditions: ...
0
votes
1answer
69 views

Show that if $\int_0^1 f(x) v(x) dx = 0$ for every function v for which $\int_0^1 v(x) dx = 0$, then f is constant. [duplicate]

Show that if $\int_0^1 f(x) v(x) dx = 0$ for every function v for which $\int_0^1 v(x) dx = 0$, then f is constant. I do not know how to do it.
2
votes
1answer
129 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
225 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
146 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
73 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
53 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 $$J(u) = \int F(u(t),\dot{u}(t)).dt, $$ ...
4
votes
2answers
253 views

fundamental lemma for variational calculus

Is it possible to use the fundamental lemma of calculus of variations in some way in the following case: $F(x,y)$ is a locally integrable function on $\mathbb{R}^n \times \mathbb{R}^n$. We know that ...
1
vote
1answer
61 views

To show that $ J(u) = \frac{1}{2}\int_\Omega |\nabla u|^2 -\frac{1}{p+1}\int_\Omega |u|^{p+1} $ is not bounded above for $1 < p < 2^*-1$

For a bounded $ \Omega\subset\mathbb{R}^n $ with smooth boundary, and for $ 1 < p < \frac{n+2}{n-2} = 2^* -1 $ where $ \frac{1}{2^*} = \frac{1}{2}-\frac{1}{n}$, I have the functional $ J : ...
3
votes
1answer
81 views

Regularity for this variational problem

The Problem. Assume $\Omega \subset \mathbb{R}^2$ bounded and $u \in H^1(\Omega,\mathbb{C})$ is some fixed function. Now consider the variational problem $$ F_\lambda(v) = \frac{\lambda}{2} ...
2
votes
1answer
163 views

Maximize a functional

Please help me how to deal with maximization of functional like this: $$F\{a(s)\} = \int\limits_0^t \left( g(a(s)) - \alpha\, v(s)^2 \right) ds, \ a(s) \in \left[0, \infty\right)$$ where $g(x) = x ...
1
vote
1answer
40 views

Restriction of a lower semi-continuous functional again lower semi-continuous?

Let $F: [a,b]\times \mathbb R \times \mathbb R \rightarrow \mathbb R$ be continuous, $J(u) = \int_{[a,b]} F(x, u, u') dx$ be a functional over $W^{k,p}([a,b])$. We assume that for any uniformly ...
1
vote
1answer
71 views

Question about $C^2$ functional

i have this problem : The solutions of P correspond to critical points of the fuctional $$\phi(x)=\frac12 \int_0^{2\pi} |x'|^2 dt - \int_0^{2\pi} F(t,x) dt , x\in E $$ where $F(t,x)=\int_0^x ...
6
votes
2answers
1k views

Conceptual difference between strong and weak formulations

What are the conceptual differences in presenting a problem in strong or weak form? For example for a 2D Poisson problem the strong form is: \begin{split}- \nabla^2 u(\pmb{x}) &= ...
2
votes
0answers
64 views

Calculus of Variations statement of a Singular Value Decomposition?

My previous question on SVD gained very little traction, so I thought I'd try a different version that hopefully has an explicit solution. As noted in the linked question, I am taking a function of ...
0
votes
0answers
63 views

Green's function

Please can someone told me how to find the Green's function $G(t,x)$ of BVP : $$u'''(t)=0 , \quad t\in (0,1)$$ and BC : $$u(0)=u'(p)=\int_q^1 w(s)u''(s) ds =0 $$ where $\frac12 < p<q<1$ are ...
3
votes
2answers
105 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 ...
1
vote
1answer
98 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 ...
1
vote
0answers
140 views

Extremal condition calculus of variations

if I have a functional with a Lagrangian $L(t,x(t),y(t),x'(t),y'(t))$, meaning two functions x and y of one parameter t. And want to solve the minimization problem $$ \int_0^t L \, dt. $$ Then I get ...