Questions tagged [variational-analysis]

Variational analysis is a branch of mathematics that extends the methods arising from the classic calculus of variations and convex analysis to more general problems of optimization theory, including topics in set-valued analysis, e.g. generalized derivatives.

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Is this “one-sided” version of the fundamental lemma of calculus of variations true?

Let $\mathbb{D}^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball, and let $A:\mathbb{D}^n \to \mathbb{R}^k \otimes \mathbb{R}^k$ be smooth. Suppose that $ \langle A , V \otimes V \...
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66 views

How to determine the path a particle that is bound to a vector field

I've been trying to solve this problem but there are no resources that help. I've tried different approaches to solve this problem but every one of them leads to a dead end. I've found one approach ...
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1answer
23 views

Existence of Geodesics

Consider a unit sphere in $\mathbb{R}^3$. How can we prove that any two points on the sphere can be joined by a minimizing geodesic? More generally, under what conditions does the same statement hold ...
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13 views

Producing term with Lie derivative from variational principle

Is it possible to obtain variational principle (Lagrangian) such that the equations of motions contain Lie derivative? For example, if $g$ is a standard metric tensor in Euclidean space $E^3$ and we ...
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1answer
29 views

Explicit Solution to Euler Lagrange Equation of Shortest Distance between two Points

So if $f(t)=(x(t),y(t))$ with $f(0)=a$ and $f(1)=b$, I should minimize $L(f)=\int_0^1{|\dot{f(t)}|}dt$. I get a jumble of equations when solving the Euler Lagrange equations with respect to $t$. Would ...
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26 views

Fundamental Lemma of Variational Calculus for Closed Surface Integrals

The fundamental lemma of the calculus of variations says that if the integral of a function times a test function over an open region is equal to zero for all test functions that vanish at boundary of ...
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1answer
32 views

Poincaré inequality for Lipschitz functions with bounded domain

Let $u\in W^{1,\infty}(B_h(0),\mathbb R^n)$, where $B_h(0)=\{x\in\mathbb R^n:|x|<h\}$. From the Poincaré inequality we know that $$ \|u-\mathrm{Id}-\frac{1}{\mathrm{Vol}(B_h(0))}\int_{B_h(0)}(u-\...
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29 views

Minimal divergence-free projection onto unit vector field

I'm attempting to use a variational method to find a critical vector field. I want to find the local components of a given vector field, $\bf{v}$, that are as "close" to $\bf{v}$ as possible, but are ...
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16 views

Optimal non-negative resampling kernel

Sampling theory says the best kernel for resampling is $sinc(x)$ which is box function in frequency domain. Which kernel is optimal under constraint that the kernel is non-negative everywhere? More ...
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1answer
31 views

Variational principal question regarding functions that have a minimum at the origin under a restriction.

I'm going over some old assignments from a couple terms ago and have come across a problem from my variational principals module. I looked at the function in the hint and noticed that for some ...
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Variational calculus … Prove that a linear functional $\varphi [h]$ cannot have an extremum unless $\varphi [h] \equiv 0$

From definitions in the book calculus of variations - Gelfand and fomin http://web.cs.iastate.edu/~cs577/handouts/variations.pdf I try prove that a linear functional $\varphi[h]$ cannot have an ...
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19 views

A variational problem involving integral

Let $W\gt 0,\,T\gt 0$ be fixed. Let $R(f)$ be an arbitraty function in $L^2(-W,W).$ Define $$\alpha_{R}:=\frac{\int_{-W}^W\,df^\prime\int_{-W}^W\,df^{\prime\prime}\frac{sin\,\pi T(f^\prime-f^{\prime\...
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25 views

Legendre Transforms

I want to show that the Legendre Transform of a function $g(x) = kf(x-x_0) + m$ for $m$, $k$ constants is equal to $g^*(p) = kf\left(\frac{p}{k}\right)+p \cdot x_0+m$ where $f^*$ is the Legendre ...
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1answer
20 views

showing a different form of the Euler Lagrange Equation provided $f\in{C^2}$ and $y'\ne{0}$

Suppose that $f(x,y(x),y'(x))$ is s.t $f\in{C^2}$ and $y'(x)\ne{0}.$ I am trying to show that the Euler-Lagrange equation $\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial y'}=0$ ...
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1answer
35 views

computing variational lower bound

I am trying to re-derive the variational lower bound for the binary logistic regression that is obtained in the paper by Saul & Jordan 1999 and it is given in equation 22 $$\langle \ln(1+e^z)\...
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2answers
41 views

formula for arc length.

Today in a variational principles lecture we were shown the following: If we have two points on the plane, namely, $(x_1,y_1)$ and $(x_2,y_2)$ and we have a curve $y(x)$ s.t. $y(x_1)=y_1$ and $y(x_2)...
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26 views

Understanding the proof of the deformation lemma

Let $V$ be a Banach space, $V\supseteq N\supseteq N_{2\rho}\supseteq N_\rho \supseteq N_\delta$. Let for every $u\in V$, $\phi_u:[0,\infty[ \to V$ be a solution of the initial value problem $$\begin{...
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28 views

Is this function differentiable

Define the following function \begin{eqnarray} f(Q_p) &=& \max \quad x^T(Q-Q_p)x +\mbox{trace}(Q_p X) \\ &&s.t. \quad (x, X) \in \mathcal{P} \end{eqnarray} where $\mathcal{P}$ is ...
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18 views

Existance of Solution in Neumann Variational Inequality

I was working on the following question and got stuck (Chap 2, #10 Kinderleher and Stampaccia) Let $\Omega$ be a bounded domain with a smooth boundary $\partial \Omega$ and suppose given $f \in H^{-...
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1answer
37 views

Find Extremum of Functional with No Initial Conditions

I have the following functional $ J(x)=\int\limits_{a}^{b}[{y(x)^2 + \dot{y}(x)^2 + 2 y(x) e^x}] dx $ and I want to find the extremum of it. So I compute the Euler-Lagrange Equation and I get the ...
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28 views

Given a smooth family of strictly convex norms in $\mathbb{R}^n$ ($n\geq 2$), what can we say about the family of dual norms?

Let $U\subset{\mathbb{R}^n}$ be an open set (may has compact closure) and $F:U\times\mathbb{R}^n\rightarrow\mathbb{R}$ such that $F(p,\cdot):\mathbb{R}^n\rightarrow\mathbb{R}$ is, for each $p\in{U}$, ...
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Minimizing elastic energy of shrinking balls

I am new to variational analysis and I am currently working in the following setup: We denote the $2$-sphere with radius $r>0$ by $S_r^2$ and $S^2:=S_1^2$. In coordinates $(\theta,\varphi)$, ...
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1answer
33 views

Calculus of Variations by Charles Fox: Question on Statement in Section 2.4

Fox states in Section 2.4, pg. 38, that "Anticipating this result, it follows that even if $u(x)$ vanishes at either or both of the values $x=a$ and $x=b$, both $t^2(a)/u(a)$ and $t^2(b)/u(b)$ still ...
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1answer
84 views

Difficulty Understanding Sufficient Conditions for Weak Extrema in Calculus of Variations

I am having a difficult time understanding Jacobi's necessary condition for weak extrema of functionals. Graphics and detailed explanations would be helpful. I am following the following two texts: ...
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31 views

Euler-Lagrange equations from a complex-valued Lagrangian

I've been looking without success for references describing a generalization of the Euler-Lagrange equations that would be derived from a complex-valued Lagrangian density. I realize that “minimum” ...
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2answers
67 views

Calculus of Variations (Gelfand & Fomin): Proof of Euler's Equation for Constrained Variation

I am in Section 2.12.1 of Calculus of Variations by Gelfand & Fomin. I am attempting to follow the proof of the Euler equation for Constrained Variation (Theorem 1, pg. 42). However, I'm confused ...
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17 views

Maximizing a nested sum of infinitely many variables (discrete variational calculus)

I am wondering if there is a way of finding a neat closed form solution for a series of numbers $r_t$ that satisfies the following. This is supposed to give the optimal strategy of an economic agent ...
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1answer
61 views

Calculus of Variations (Gelfand & Fomin): Proof of Functional Dependence Identity

I'm on page 40 of the book (Section 2.10 - Variational Problems in Parametric Form). It states that for $\int_{t_0}^{t_1}F\left(x,y,\frac{\dot{y}}{\dot{x}}\right)\dot{x}dt=\int_{t_0}^{t_1}\Phi\left(x,...
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45 views

Is the Clarke Subdifferential always defined for Lipschitz continuous functions?

According to various websites, for some function $f:X\to R$ we can define a map $$ D(x,v):= \lim_{y\to x} \sup_{t\searrow0} \frac{f(y+tv)-f(y)}{t} $$ and the Clarke Subdifferential (Def 1) is $$ \...
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1answer
34 views

Interior solution of a Variational Inequality in Hilbert Space

I'm trying to understand the proof of the following claim: Let $K \subset \mathbb{R}^n$ be compact and convex and let $$F:K \rightarrow (\mathbb{R}^n)'$$ be continuous. Then there exists an $x\in K$...
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2answers
56 views

Using Hamilton's principle to derive Newton's equations of motion in parabolic coordinates

I have recieved a very hard (optional) assignment on variational calculus, and I have not got a clue where to start other then stating the Euler-Lagrange equations. Here is the problem: According to ...
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1answer
41 views

The explicit expression of $\frac{δF}{δP}$

I'm writing simulation code of ferroelectric domain, and there is a math problem that I can't solve. The expression of $F$ is $$ F = \frac{|\vec{k} \cdot \vec{P}(\vec{k})|^2}{k^2}. $$ $\vec{k}$ is a ...
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1answer
59 views

Why is the epigraph of Moreau-Yosida Regularization a projection of a convex set?

The Moreau-Yosida Regularization is given by \begin{equation} f_\mu(x) = \inf_y \left( f(y) + \frac{1}{2\mu} \| x - y \|^2 \right). \end{equation} We know that $L(x, y) = f(y) + \frac{1}{2\mu} \| x -...
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16 views

Graphical Convergence

Suppose that $U$ is a dense subset of $\mathbb{R}^d$, what is an example of a lower semi-continuous map $f:\mathbb{R}^d\rightarrow \mathbb{R}$, for which $f|_U$ converges graphically to $f$? Note: ...
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2answers
159 views

Courant (1943) and History of Finite Element Method

I am interested in the history of Finite Element Methods and Methods of Weighted Residuals (MWR), especially reduced quadrature and collocation methods. I have a paper coming out called “Orthogonal ...
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20 views

Guaranteeing isoperimetry constraint for non-extremal functional in PDE.

First of all, hello and thank you for your time. Context I am making a program that solves the differential equation for the time evolution of a system from the equations: $$F[\mathbf{y}]=\int\...
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1answer
49 views

Determining a variational formulation of $u^{(4)} = f$ with $3$-rd order BC.

Below is a problem from a recent exam and I have some questions about it. Given the boundary value problem:$$\dfrac{\partial^4 u}{\partial x^4} = f,\quad f\in L^2(0,1)$$ $$u(0) = u''(0) = u'(1) = u''...
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1answer
65 views

With the solution of the Euler-Lagrange equation prove the following equation

Let $\Omega \subset \mathbb{R}^n$ and $F=F(z,p):\Omega \times \mathbb{R} \times \mathbb{R}^n$ be smooth and independent of $x\in\Omega$. Let $u$ be a solution of the Euler-Lagrange equation of $\...
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43 views

Weak derivative of $u(x)=|x|$ not belong in $W^{1,p}(-1,1)$

Consider the functión $u \in W^{1,p}(-1,1) $, defined by $u(x)=|x|$, we know its weak derivative is $$g(x)=\left \{ \begin{matrix} 1 & \text{if }x\in(0,1) \\ -1 & \text{if } x \in (-1,0) \...
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44 views

Does functional satisfies Palais-Smale condition?

Check if the functional $f(u)=\int_0^{1/2} u^2(x)dx$ satisfies the Palais-Smale condition on the Hilbert space $L^2([0,1],\mathbb{R})$. We have definied the Palais-Smale condition as follows: $f$ ...
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73 views

How to treat an equation of the form $-\Delta u=G\cdot \nabla u+f(u) ?$

There are plenty of variational techniques (direct methods of calculus of variations, mountain pass type theorems, Lusternik-Schnirelmann theory) to prove the existence of solutions of a semilinear ...
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52 views

How is the Lagrangian function homogeneous in the velocities?

I'm working through The Variational Principles of Mechanics by Cornelius Lanczos and in chapter 6, section 10, he says that the Lagrangian function is homogeneous in the velocities. $L_1 = L(q_1,......
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30 views

Minimum expected value over all probability functions

Find the minimum value of $E[X]$ over all probability density functions f(x) satisfying the following three constraints: (I) $f(x) = 0 $ for $x \leq 0$ (II) $ \int_{-\infty}^{\infty} f(x) dx = 1 $ ...
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1answer
52 views

Neumann problem with zero average sobolev space

Let us considère the following Laplace-Neumann problem $-\Delta u=0$ with homogenuous boundary condition of type neumann, i:e $\frac{{\partial u}}{{\partial n}} = 0$. The variational formulation is ...
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1answer
48 views

How to take derivative of integral of function?

I'm reading a textbook where it forms a Lagrangian function $$ L = \int_0^1 f(x)^{1 - \frac{1}{\alpha}}dx - \lambda\int_0^1 g(x) f(x) dx$$ But how do you take the derivative of this thing? The ...
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89 views

Optimization in Banach space: Find functions that minimize the supremum of a convex operator.

Let $D \subset \mathbb{R}^n$ be compact. Denote by $C(D, \mathbb{R}^n)$ the space of continuous functions from $D$ to $\mathbb{R}^n$. Let $K$ be a real, symmetric, positive-definite $n \times n$ ...
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37 views

Intuition behind Pohozaev identity

I'm wondering if the Pohozaev identity: $$n\int_\Omega\int_0^{u(x)}f(t)\operatorname{d}t\operatorname{d}x-\frac{n-2}{2}\int_\Omega u(x)f(u(x))\operatorname{d}x=\frac{1}{2}\int_{\partial\Omega}\left|\...
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1answer
328 views

Confused by Kullback-Leibler on conditional probability distributions

I understand the Kullback-Leibler divergence well enough when it comes to a probability distribution over a single variable. However, I'm currently trying to teach myself variational methods and the ...
4
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96 views

Does this functional satisfies the Palais-Smale condition?

Let $\Omega$ be a non-empty bounded open subset of $\mathbb{R}^N$, $\lambda\in \mathbb{R}$ be an eigenvalue of $-\Delta$ on the Sobolev space $H^1_0(\Omega)$ and $f\in L^\infty(\Omega\times\mathbb{R})$...
3
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1answer
82 views

Characterisation of the rate distortion function: issue with functional derivative

In Elements of Information Theory, I can't figure out how the functional derivative $ \frac{\delta J}{\delta q(\hat{x}|x)} $ for $ J(q) = \sum_x \sum_{\hat{x}} p(x)q(\hat{x}|x)\log{\frac{q(\hat{x}|x)}{...