0
votes
0answers
21 views

Calculus of variations with $y\ge 0$ constraint [closed]

In my problem of Calculus of variations $J(y)=\int_a^b L(x,y,y')dx=0$, I have : end point constraint; integral constraint; and $y\ge 0$. We are familiar with the first two constraints, but what ...
0
votes
1answer
15 views

Strong and weak extrema

I am confused about the "strength" of the two definitions. The definitions I use are the following: Let $y$ be a function defined on the set $M$. Neighborhood (0. order) of the function $y$ is the ...
0
votes
1answer
44 views

Are all definite integrals considered functionals?

In my Optimization class, we are messing around with some Calculus of Variations in an effort to find functions which minimize functionals. In these cases, the spaces we're working with are spaces ...
0
votes
1answer
26 views

Prove that solution of a variational problem exists

Let $X$ is a Hilbert Spaces We define two operators $$a:X\times X\rightarrow\mathbb R$$ and $$b:X\rightarrow\mathbb R$$ where $a$ is a symmetric, bounded, strongly positive operator, and $b$ is a ...
2
votes
1answer
40 views

How to calculate this functional derivative?

How can I calculate the functional derivative of this functional? $$F[x](t) = \int_{0}^{t}x(t_1)a(t_1)\left \{ \int_{0}^{t_1}x(t_2)b(t_2) \,dt_2\right \} dt_1 .$$ Where $a(t)$ and $b(t)$ are real ...
0
votes
0answers
50 views

Open problems in variational analysis/PDEs

I wasn't sure whether this question was more appropriate for StackExchange or Overflow, but in any case I would really appreciate it if any active researchers in the field responded. I'm a PhD ...
0
votes
0answers
39 views

Fréchet normal cone

Given $x\in \Omega(\subset X)$ (X: Banach space) and $\varepsilon\geq 0$, the set of $\varepsilon-$normals to $\Omega$ at $x$ by \begin{align} \widehat N_\varepsilon(x;\Omega):=\left\{x^*\in X^*\mid ...
2
votes
1answer
36 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
55 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
72 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
68 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
28 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
28 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
45 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
16 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
66 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
26 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
26 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
41 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
23 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
44 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
91 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
64 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
79 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
58 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
60 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 ...
3
votes
1answer
127 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
154 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
38 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
49 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
76 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
113 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
112 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
134 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
231 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
2answers
199 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
82 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
55 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
58 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
272 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
67 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
82 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
205 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
47 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 ...