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

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Non-linear hyperbolic systems

This question is about a naive approach to non-linear hyperbolic systems, thinking in the context of elasticity. To set up the problem suppose $\Omega\subset \mathbb{R}^n$ is open and bounded. ...
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1answer
20 views

Euler equation formula

When I am using Euler equation for Fourier transform integrals of type $$\int_{-\infty}^{\infty} dx f(x) exp[ikx] $$ I am getting following integrals: $\int_{-\infty}^{\infty} dx f(x) cos(kx)$ ...
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33 views

Fundamental Lemma of the Calculus of Variations with higher derivatives

The fundamental lemma of the calculus of variations is often presented as: If $M(x) \in C[a,b]$ such that $\int_{a}^{b}{M(x)\eta(x)} = 0 ~~\forall\eta\in C^1[a,b],\eta(a)=\eta(b)=0$, then $M(x)=0$ for ...
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19 views

Differentiate functional with delta function when calculating Euler-Lagrange equation

The paper "active contours without edges" by Chan and Vese http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=902291, My goal is to understand how to derive the corresponding euler-lagrange ...
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1answer
24 views

The meanings of some symbols in “Calculus of variations”

Could someone tell me the meanings of the "C" and its superscript "1" and subscript "0" in the equation which I have marked. Thank you very much!!!
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35 views

What curved ramp transports a ball from (1,1) to (0,0) most quickly, under the acceleration of gravity, with no friction or air resistance?

An infinitisemally small ball is placed at the top of a ramp which has a height of 1m and ends 1m away horizontally. What is the optimal curve of the ramp to minimize time taken for the ball to reach ...
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1answer
25 views

calculus of the measure of a $C^1 $ hypersurface

I have to prove that: $$\lim_{r \to 0} \frac {\mathcal H ^{n-1}(M \cap B(x,r))}{\omega_{n-1} r^{n-1}}=1, $$ where $\mathcal H ^{n-1}$ is the ($n-1$)-dimensional Hausdorff measure, $M$ is a $C^1$ ...
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70 views

A variational approach to symplectomorphisms

Let $(\Sigma,\omega)$ be a compact symplectic surface, and let $\mathcal{D}$ be the group of diffeomorphisms of $\Sigma$. Is there some known variational approach to determine if an element ...
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1answer
23 views

Absolute Continuity defined by Necas

I read a definition of the absolute continuity in Necas' book "Direct Methods in the Theory of Elliptic Equations": Let $\Omega$ be a domain in $\mathbb{R}^n$ , $P$ a line verifying ...
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1answer
33 views

Elementary Examples of Functionals

I'm working on a research project that's a little over my head, so forgive some simple questions. Is a composite function $f(g(x))$ a functional? What are a handful of other simple examples of ...
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1answer
16 views

Regularity of limit measure and prove that $|\mu_h|\stackrel{*}{\rightharpoonup}|\mu|$

I have some questions. First of all, let $\mu_h$ a sequence of Radon measures and suppose that $\mu_h$ weakly-converge to another measure $\mu$. Now, this limit measure $\mu$ is still Borel? Is it ...
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1answer
45 views

How to Invert the Euler Lagrange Equations?

Suppose I have a functional L. For example $L = y+3y'$. Where y is itself a function of real variable x It's easy for me to evaluate the Functional Derivative of L via the Euler Lagrange Equations: ...
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2answers
28 views

Minimum curve for the distance between two points at the plane

The problem is to determine the curve y=y(x) in the plane, the lenght of which is given by the functional: \begin{equation} I(y)=\int_{x_1}^{x_2}\sqrt{1+(y')^2}dx=\int_{x_1}^{x_2}F(x,y,y')dx ...
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1answer
21 views

Calculus of Variation: Euler-Lagrange Equation in 1D

I am currently trying to get into calculus of variation for a course of Image processing. In the lecture we learned that a smooth function u[a,b]->R that minimises: $$\int_a^b F(x,u,u') dx $$ ...
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1answer
16 views

Prove that a sequence of measures weak-star converges to another measure

We have a set of locally finite perimeter and a sequence of sets $\{E_h\}_h$ with $C^1$ boundary such that $$E_h\to E \text{ and } \mu_{E_h}\stackrel{*}{\rightharpoonup} \mu_E,$$ where $\mu_{E_h}$ and ...
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41 views

The Dirichlet problem for the Laplace equation: classical solutions versus weak solution

Let $B_R$ a ball in $\mathbb{R}^n$. Consider $u^{\star} \in H^{1}(B_R) $ and $f \in H^{1}(B_R) \cap C(\overline{B_R})$. Suppose that $u^{\star}$ minimizes $$\int_{B_R} |\nabla u|^2, u \in \{ v \in ...
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How to use find the Lagrange Multipliers in multidimensional Calculus of Variations

Suppose I wish to minimise the integral $$I = \int_{s_0}^{s_1}\int_{t_0}^{t_1}F\, dt ds$$ Where $F$ is a function of the six variables $x(s,t)$, $y(s,t)$, and their four partial derivatives, ie $$F ...
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weakly lower semi contiuous functional

Let $\Omega$ be a bounded open domain of $\mathbb R^n$, if we have the functional defined on $W^{2, p}_0(\Omega)$ by $$F(w) = \frac 1p \int_\Omega |D^2 w|^p dx.$$ I read in some papers this is a ...
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1answer
45 views

Euler-Lagrange equation with constraints outside the integral

So I've been studying Euler-Lagrange equations, and on an assignment I have the problem to find them for $J(y)=\int_a^bF(x,y,y')dx-By(b)+Ay(a)$ Where $y(a)$ and $y(b)$ are free, $A$ and $B$ are ...
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1answer
69 views

Surface of constant mean curvature

From PDE Evans, 2nd edition: Chapter 8, Exercise 12: Assume $u$ is a smooth minimizer of the area integral $$I[w]=\int_U (1+|Dw|^2)^{1/2} \, dx,$$ subject to given boundary conditions $w=g$ on ...
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0answers
12 views

Definition of the Second Variational Derivative In terms of The first

I know that for functional $F$ the first variational derivative at $f$ with increment $h$ is defined as \begin{align*} \delta F[f,h]= \lim_{\alpha \to 0 } \frac{F[f+\alpha h]-F[f] ]}{\alpha }. ...
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2answers
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dirichlet principle: why $u-g\in W_0^{1,2}(\Omega)$?

Let $D\subset\mathbb R^n$ be open and bounded. Consider $\Delta u=f$ in $\Omega$ and $u=g$ on $\partial\Omega$. Let $g\in W^{1,2}(D)$ and $f\in L^\infty(D)$. Then the minimizer of $$ I(u)=\int_\Omega ...
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Calculus of variations - unilateral constraints [duplicate]

Question about Evans states, chapter 8.4.2! We have $I[w] := \int_U \frac{1}{2}|Dw|^2 - fw\, dx$, among all functions $w$ belonging to the set $$\mathcal{A} : = \{w \in H_0^1(U) : w \geq h \, ...
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1answer
51 views

The functional $I[w] = \int_U \frac{1}{2} |Dw|^2 - fw \, dx$ is weakly lower semicontinuous

I am studying calculus of variation, and I need to prove that $I[w] = \int_U \frac{1}{2} |Dw|^2 - fw \, dx$ with $f \in L^2(U)$ is weakly lower semicontinuous on $H_0^1(U)$. In classes, I only ...
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1answer
15 views

Euler equation for the functional has the form: $f_y-f_xy'-\frac{fy''}{1+y'^2}=0$

I want to show that the Euler equation for the functional $J(y)= \int_a^b f(x,y) \sqrt{1+y'^2}dx$ has the form: $$f_y-f_xy'-\frac{fy''}{1+y'^2}=0$$ $$L(x,y,y')= f(x,y) \sqrt{1+y'^2} dx$$ ...
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1answer
27 views

optical flow Euler-Langrange equation

I have a problem understanding how optical flow functional is plugged into Euler-Lagrange equation. The functional is: $\iint[(I_xu+I_yv+I_t)^2+\alpha^2(||\nabla u||^2 +||\nabla v ||^2)]dxdy$ ...
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Geodesic equation for surface of sphere

One of the standard problems of calculus of variations is showing that geodesics on the surface of the sphere are great circles. But I don't understand the equation. The equation for great circle ...
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1answer
30 views

Dirichlet energy

From PDE Evans, 2nd edition: Chapter 8, Exercise 17: Let $u,\hat{u} \in H_0^1(U)$ both be positive minimizers of the Dirichlet energy $$I[w] := \int_U |Dw|^2 \, dx,$$ subject to the constraint ...
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2answers
223 views

Elliptic regularization of the heat equation

This is from PDE Evans, 2nd edition: Chapter 8, Exercise 3: The elliptic regularization of the heat equation is the PDE $$ u_t - \Delta u -\epsilon u_{tt}=0 \quad \text{in }U_T, \tag{$*$}$$ where ...
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0answers
39 views

Derivation of Von Karman Equations

I'm reading Howell's Applied Solid Mechanics to gain background for a research project. I'm struggling with the following derivation in the text that the authors refer to as a "lengthy exercise." The ...
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1answer
94 views

Euler-Lagrange equation [duplicate]

This is PDE Evans, 2nd edition: Chapter 8, Exercise 2: Find $L=L(p,z,x)$ so that the PDE $$-\Delta u + D\phi \cdot Du = f \quad \text{in }U$$ is the Euler-Lagrange equation corresponding to the ...
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1answer
46 views

Calculus of Variations - Function dependent of $y'$ only

I completely understand the proof for the Euler-Lagrange equation for a general function $F(x,y,y')$. However, when I try to use the same proof technique on a function $F(y')$, I run into a curiosity ...
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2answers
228 views

Maximum area of a fenced playpen on the side of a house.

Here's an interesting problem: you just got a really cute puppy, and you want it to have a large rectangular playpen to run around in. What's more, your neighbor just happened to have 100 feet of ...
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Calculus of variational in 2 dimension with constraints

Let $S$ be a 2D region and its boundary is $\partial S$. $u(x,y)$ is defined in S. The functional is of the following type: $J[u] = \int_S F(x,y,u,u_{x},u_{y}) \mathrm{d}s + \int_{\partial S} ...
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1answer
135 views

What function maximizes area for a constant arc length?

Suppose I have a continuous function $f$, such that $f(0) = f(1) = 0$. Given the length $l$ of the curve between $0$ and $1$, which function maximizes the area under the curve? I know that if $l \leq ...
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0answers
59 views

How to prove $H_0^1(\Omega)=H_0(div;\Omega)\cap H_0(curl;\Omega)$

It was shown in the book "Finite Elements Methods for Navier-Stokes Equations" by Girault and Raviart that $$H_0^1(\Omega)=H_0(\operatorname{div};\Omega)\cap H_0(\operatorname{curl};\Omega).$$ The ...
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0answers
43 views

Find extremum of functional

I want to find the extremum of $$J(y)= \int_1^2 \frac{\sqrt{1+y'^2}}{x}dx, \ y(1)=0, \ \ y(2)=1$$ I thought to use the following theorem: If $y$ is a local extremum for the functional $J(y)= ...
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1answer
49 views

Why does the functional have a local minimum at $0$?

Definition: Let $J: A \to \mathbb{R}$ be a functional , where $A \subset V$ and $(V, ||\cdot||)$ a linear space with norm. Let $y_0 \in A$ and $h \in V$ such that $y_0+ \epsilon h \in A $ for ...
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1answer
12 views

Minimizing constrained functions on $l^p$

Suppose we have some functionals $H,G:l^p(\mathbb{N}^+)\to\mathbb{R}$, and we want to find some $p \in l^p(\mathbb{N^+})$ which minimize $H$, subject to the constraint that $G(p)=0$ is constant. As ...
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25 views

An MCQ involving Rayleigh - Ritz method for the functional $I(y) = \int_{0}^{1}(\frac{1}{2}(y^{'})^2 - y)dx$

An MCQ involving Rayleigh - Ritz method for the functional $$I(y) = \int_{0}^{1}(\frac{1}{2}(y^{'})^2 - y)dx$$ Let $y_\text{app}$ be polynomial approximation, involving only one coordinate ...
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1answer
13 views

An MCQ for finding the extremal of the functional $J = \int_{a}^{b} F(x, y, y^{'})$

Consider a functional $$J = \int_{a}^{b} F(x, y, y^{'}),$$ where $F(x, y, y^{'}) = \frac{1 + y^{2}}{(y^{'})^2}$ for admissible function $y(x).$ Which of the following are extremals for $J$? $y(x) = ...
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1answer
54 views

Intuition of weak star convergence.

Given $\Omega=(0,1)$, consider the following sequence $$ v_j(x)\colon=\begin{cases} \;a &\text{if }jx-\lfloor jx \rfloor\le\theta\\ \;b &\text{otherwise} \end{cases} $$ where ...
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1answer
72 views

Functional Maximization

So how do we solve a problem like this: Find the function $s(x)$ such that $s(x)$ maximizes $$\int_0^{s^{-1}(k)} s(x) dx $$ where $x\in[0,10]$, $s(x)\in[0,1]$, and $k\in[0,1]$ ($k$ is a constant). ...
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0answers
36 views

Ideal shape for underwater habitat

Is there an analytic solution to this problem or do I need to compute a discrete approximation using a relaxation procedure - or something similar? I want to find the shape of a roughly spherical ...
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1answer
33 views

Calculate the (variational) derivative of the following equation;

Consider $ E[u]= \int^1_0 \big(u'(x)\big)^2+\big(u(x)\big)^2-2f(x)u(x) dx.$ Calculate the variational derivation for a function $v$; in other words, calculate $\frac{d}{d\epsilon}E[u+\epsilon v]$ at ...
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0answers
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An MCQ to determine extremal of a functional $J(y) = y^2(1) + \int_{0}^{1}(y^{'})^2(x)dx,$ [duplicate]

The following problem occurs in an exam: Consider a functional $J(y) = y^2(1) + \int_{0}^{1}(y^{'})^2(x)dx,$ $y(0) = 1, $ where $y \in C^2 ([0, 1]).$ If $y$ extremizes $J$, then $y(x) = 1 ...
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2answers
68 views

Advice on second order non-linear ordinary differential equation

I'm currently working on some problems concerning the calculus of variations and I have come up with the following differential equation that I now want to solve: $$1 + y'(x)^2 - y''(x)(y(x)-\lambda) ...
2
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1answer
50 views

Divergence identity

From PDE Evans (2nd edition), page 515, we are given $$\sum_{i=1}^n \left(\left(Du \cdot x + \frac{n-p}p u \right)p|Du|^{p-2}u_{x_i}-|Du|^px_i \right)_{x_i}=0. \tag{10}$$ Then the author goes ...
2
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2answers
52 views

Lagrangians independent of $x$

In PDE Evans, 2nd edition, the following formula is printed as equation $\text{(9)}$ in §8.6 (on page 514): $$\sum_{k=1}^n (L_{p_i}u_{x_k}-L\delta_{ik})_{x_i}=0 \quad (k=1,\ldots,n) \tag{9}$$ ...
2
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2answers
79 views

Minimizing a functional with a free boundary condition

Find the extremals of the functional $$\text{J}(y)= y^2(1) + \int_0^1 y'^2(x)dx , \ \ y(0)=1.$$ Answer: $y(x)=1-\frac{1}{2}x$ My solution: $ F (x,y,y')=y'^2(x)$ After solving the ...