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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1answer
130 views

If $f$ is proper, lsc, and $\frac{f(x) + f(y)}{2} = f^{**}\left(\frac{x + y}{2}\right) \implies x = y$, is $f$ necessarily convex?

Suppose $X$ is a real Hilbert Space and $f : X \to (-\infty, \infty]$ is a lower semicontinuous, proper function. Further, suppose $f$ satisfies the following, for all $x, y \in \operatorname{dom} f$: ...
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1answer
146 views

Do there exist energy-minimizing immersions?

Let $M,N$ be $d$-dimensional connected oriented Riemannian manifolds, possibly with boundary, $M$ compact. Let $E_d:C^{\infty}(M,N) \to \mathbb{R}$ be the $d$-energy, i.e. $$ E_d(f)=\int_M |df|^d \...
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108 views

Minimal Surface and Mean Curvature

In the frame of a course on instabilities, I am trying to prove that the soap film between two rings has the form of a catenoid. Since pressure is equal on either side of the film, we expect to have a ...
5
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44 views

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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315 views

Looking for a rigorous treatment of the functional derivative the way it's used by physicists.

A lot of theories in physics can be derived from a variational principle: some action functional S on a space of e.g. field configurations $\phi$ is given, and the equations of the theory follow from ...
5
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123 views

All the symmetries of the Dirichlet energy are conformal

It seems to be "folklore" knowledge that all the (source) symmetries of the $d$-Dirichlet energy are conformal maps. Specifically, I have found this nice proof for the following claim: Proposition:...
4
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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 ...
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 $f\in L^\infty(\Omega\times\mathbb{R})$...
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61 views

Equivalent variational formula for eigenvalues

First we define Rayleigh quotient as $$R(u)=\frac{\int_M|\nabla u|^2 dV_g}{\int_M |u|^2d V_g}$$ We can deduce that the first nonzero eigenvalue of Laplacian operator is $$\lambda_1=\inf\{R(u): \int_M ...
3
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1answer
419 views

Finding extremal of the functional

Find the stationary point of the functional $$ J[y]=\int \left( x^2y'^2+2y^2 \right) dx $$ where $y(0)=0, y(1)=2.$ My Solution: E-L equation: $x^2y''+2xy'-2y=0.$ This is also Cauchy-Euler equation. ...
3
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1answer
107 views

Let $X$ be dense in $Y$. Looking for an example such that a bounded bilinear form fulfills an inf-sup condition on $X\times Y$ but not on $X\times X$

Let $X$ and $Y$ be two separable Hilbert spaces such that $X \subsetneq Y$ densely, continuously and the norms are related through $\|x\|_X^2=|x|_X^2+\|x\|_Y^2$ for all $x \in X$; e.g. take $X=H^1$, $...
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)}{...
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1answer
112 views

A mistake on the Borwein - Preiss's Variational Principle statement??

I know the following generalization of Borwein - Preiss's Variational Principle(BPVP), known as Loewen-Wang's Variational principle (LWVP) $\textbf{Loewen- Wang Variational Principle}$ Let $...
3
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1answer
33 views

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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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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36 views

Lagrange Multipliers for Bingham Flow

I want to prove existence and uniqueness of Lagrange multipliers of the following variational inequality (which is derived from Bingham fluids) \begin{equation*} \int_\Omega \nabla u \cdot \nabla (v-u)...
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70 views

Variational Equation in Perturbation Theory Chapter of Arnold *Geometric Methods in ODE*

In the preface to Chapter 4 ``Perturbation Theory" in Arnold's book Geometric Methods in the Theory of Ordinary Differential Equations available for preview here https://www.springer.com/gp/book/...
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90 views

Convexity vs Quasi-convexity

It is well known that in optimization the concept of quasi-convex function is the following: $f: \Omega\subset \mathbb{R}^n \rightarrow \mathbb{R}$ is a quasi convex function if for every $\alpha\in\...
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2answers
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$0\leq f,g\in C^{1},\int_{a}^{b}\sqrt{f}\geq\int_{a}^{b}\sqrt{g}\implies\int_a^b f\geq\int_{a}^{b}g$?

$f,g$ are differentiable non-negative functions on $[a,b]$ with $ \int_{a}^{b}\sqrt{f(t)}dt\geq\int_{a}^{b}\sqrt{g(t)}dt $. So do we have that $\int_{a}^{b}f(t)dt\geq\int_{a}^{b}g(t)dt$ ? Does ...
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1answer
53 views

Why minimize $ \Vert u_1 + u \Vert_1 $ in this Finite Element Analysis variational problem?

$\textbf{The problem reads:}$ Let $\Omega$ be bounded with $\Gamma: = \partial \Omega$ and let $g:\Gamma \rightarrow \mathbb{R} $ be a given function. Find the function $u\in H^1(\Omega)$ with ...
2
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1answer
54 views

An example for a stable harmonic map which is not a local minimizer

I am looking for an example of a harmonic mapping between compact Riemannian manifolds, which is stable but not a local minimizer of the Dirichlet energy. A harmonic map $f:M \to N$ is said to be ...
2
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1answer
236 views

$\epsilon$-normals to convex sets

I am reading the book by B. Mordukhovich, Variational analysis and generalized differentiation I. On page 6 it is stated the following inclusion: $$ \hat{N}_{\varepsilon }\left( \bar{x};\Omega \right)...
2
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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 ...
2
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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 ...
2
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2answers
313 views

Permutation and combinations using chairs?

After reading and watching a lot about permutation, combination and variation i still don't understand them fully. So i have two questions: How many ways are there to position 5 people on 10 chairs? ...
2
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1answer
48 views

Finding consumption function which maximizes utility

I can across this question in my applied real analysis textbook that I'm having trouble with. It asks us to consider the utility function $U(C) = \sqrt{e^{-rt}C}$. I'm supposed to find the consumption ...
2
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3answers
137 views

Euler-Lagrange formula

Let $y:[-1,1]\to [2-1,2+1]$ be a $C^1$-smooth function, and $F(y,y'):=y\sqrt{1+y'^2}$. Suppose $y(x)$ satisfy the Euler-Lagrange equation, i.e. $$\frac{\partial F}{\partial y}-\frac{d}{dx}\frac{\...
2
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0answers
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 ...
2
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0answers
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 $$ \...
2
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0answers
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 $n \times n$ ...
2
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1answer
58 views

Simple exercise with differentials

Given the diferentials $$ \begin{equation} d'Q=K(x-x')d'x \end{equation} $$ $$ \begin{equation} d'Q'=K(x'-x)d'x' \end{equation} $$ where $K$ is a constant, I need to show that $$ \begin{equation} \...
2
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0answers
77 views

Understanding Arnold's definition of “differentiable”

I'm looking at Arnold's Mathematical Methods of Classical Mechanics at the beginning of chapter 3 p 55 which defines when a functional is differentiable. Slightly paraphrasing and skipping a couple ...
2
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1answer
50 views

Mean field approximation

I am a Japanese student studying machine learning and stuff like that. When I have studied Mean Field Approximation HERE, I got question. In that post, equation (14) is as follows: $$ L[q_1,...,q_m] -...
2
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0answers
50 views

Is there always a non-trivial homotopy class of maps which minimizes the Dirichlet energy?

Let $M,N$ be $d$-dimensional oriented Riemannian manifolds, possibly with boundary, $M$ compact. Let $E:C^{\infty}(M,N) \to \mathbb{R}$ be the Dirichlet energy, i.e. $$ E(f)=\int_M |df|^2 \text{Vol}...
2
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0answers
75 views

Functional form of an Ordinary Differential Equation

I was learning about Variational Method to solve the differential equation for the following differential equation: $$\frac{d^2u}{dx^2}-u=-x, 0<x<1$$ $$u(0)=0, u(1)=0$$ However, the textbook I ...
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0answers
69 views

Solving $\frac{d^2 \theta}{d x^2} - m^2\theta = 0$ using the Ritz method

I'm trying to solve the following ODE using the Ritz method: $$\frac{d^2 \theta}{d x^2} - m^2\theta = 0$$ With the boundary conditions $$\frac{d\theta}{dx}\Bigg{|}_{x=0} = 0$$ $$\theta(1) = \...
2
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0answers
34 views

variation method

Define $ F[\phi]=\int dr \{V(\phi)+\frac{1}{2} \kappa \vert \nabla \phi \vert^2 \}$, then how can we deduce that$ \frac{\delta F}{\delta \phi}= \frac{d V}{d \phi}-\kappa \nabla^2 \phi $? I feel ...
2
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1answer
102 views

The integral under the variation $\delta$ sign

In physics books on classical field theory, the authors usually define the action as $$ S = \int\mathcal{L(\phi,\partial_\mu\phi)d^4x} $$ where $\mathcal{L}$ is the lagrangian density. Then, ...
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1answer
781 views

What are prerequisites to Terry Tao's An Introduction To Measure Theory?

I am an economics student and want to study mathematics, variational analysis in particular, with measure approach but since I am ignorant of measure theory I decided to try this book but I still find ...
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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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1answer
127 views

what is a variational form in optimization?

This is probably a pretty basic question, but I can't figure out what people mean when they say "variational forms" in optimization. For example, in this paper I'm reading, the variational form of a ...
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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
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 $\...
1
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1answer
70 views

Is the normal cone well defined?

I am reading Mordukhovich's Variational Analysis and Generalized Differentiation, and the first definition there is that of a $\epsilon-$normals Given a Banach space $X,$ and $\Omega\subseteq X,$ ...
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1answer
593 views

Solving PDEs using the Ritz method on variational calculus problem (Student questions)

I'm reading the book "Conduction Heat Transfer" by Vedat S. Arpaci. I'm currently at chapter 8 (I didn't read the rest of the book, though), which talks about the Variational Formulation - Solution by ...
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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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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)...
1
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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 ...
1
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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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1answer
55 views

Wondering how to get this analytical solution of $\text{E}\big(\log(f)\big)$, $f\sim$ Normal Distribution

I am reading variational inference for gaussian process modulated poisson processes and find the result (19) is unclear about its source. I am wondering how they get that. The equation is shown here \...