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
152 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 $x, y \in \operatorname{dom} f$: ...
7
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140 views

How to find $\operatorname{argmin}_{\int_{\Omega}\Delta u=0, u(z_1)=u_1,…,u(z_m)=u_m}{\|\Delta u\|}$?

Let $d\in\mathbb{N}$ and let $\Omega$ be a non-empty bounded arcwise connected open subset of $\mathbb{R}^d$ with regular boundary. Denote by $C^1(\bar\Omega)$ the set of differentiable continuous ...
6
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1answer
156 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 \...
6
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0answers
109 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 $f\in L^\infty(\Omega\times\mathbb{R})$...
5
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2answers
180 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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0answers
50 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 $(\theta,\varphi)$, ...
5
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387 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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0answers
150 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
1k 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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2answers
318 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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1answer
49 views

Calculate the first variation of $\int_{a}^{b} \sqrt{1+|\frac{dy}{dx}|^2}~dx$ by considering a parameterisation of y and x

So i've been given this job, i could use some help please. This is what i have so far: $$J(x,y,y') = \int_{a}^{b} \sqrt{1+\left|\frac{dy}{dx}\right|^{2}} dx \implies $$ $$J(t,x,y,\dot{x},\dot{y}) = \...
4
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69 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
763 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
113 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
90 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)}{...
3
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1answer
117 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 $f:X\to \...
3
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1answer
44 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 \...
3
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0answers
83 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 ...
3
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39 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)...
3
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0answers
130 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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97 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\...
2
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2answers
101 views

$0\leq f,g\in C^{1},\int_{a}^{b}\sqrt{f}\geq\int_{a}^{b}\sqrt{g}\implies\int_a^b f\geq\int_{a}^{b}g$?

$f,g$ 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 ...
2
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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
60 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
votes
1answer
28 views

Where is my error when setting up the differential equation to find the length of a path on a sphere

I have a sphere and I want to find the equation of the curves which give us the shortest path between two points laying on its surface. By using the Euler-Lagrange equation, I need to find the value ...
2
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1answer
88 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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1answer
259 views

$\epsilon$-normals to convex sets

I am reading the book Variational Analysis and Generalized Differentiation I by B. Mordukhovich. On page 6 it is stated that the following inclusion: $$ \hat{N}_{\varepsilon }\left( \bar{x};\Omega \...
2
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2answers
62 views

Euler-lagrange equation solution doesn't make sense

I have a real world based optimization problem, where the equation is $$ T=\int_{0}^{A}\frac{\sqrt{1+y'(x)^2}}{v(x)}\mathrm{d}x,\tag{1}$$ so that $$ L_{y'}=\frac{y'(x)}{\sqrt{1+y'(x)^2}\cdot v(x)}, \...
2
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1answer
50 views

Calculus of variation; Calculating First Variation

so from my understanding of the subject there seems to be a whole deluge of differing definitions for things such as the First variation for a functional. now i've been asked to calculate the first ...
2
votes
1answer
66 views

Using Lagrange multiplier in Euler-Lagrange Equation

I think I am doing something wrong when combining Lagrange multiplier and Euler-Lagrange equation. I need to maximize a functional of the form: $$ \int\!dx~{L(x, G, \dot{G})}~~~~~\text{where } L(a, b,...
2
votes
1answer
109 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
votes
1answer
37 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
691 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
52 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
votes
3answers
153 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
40 views

Euler Lagrange equation in variational calculus for a sum of integrals

Let $F, G: \mathbb{R}^3\rightarrow{}\mathbb{R}$ be two continuously differentiable functions and let $a\leq b \leq c$. I want to know if there exists some known method to find a function that ...
2
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1answer
52 views

Integration by parts with cross derivatives

I wish to solve the following simplified problem in the context of Weak Formulations $\large \iint(u \frac{\partial ^2 v}{\partial x ^2})dxdy + \iint(u \frac{\partial ^2 v}{\partial x \partial y})dxdy ...
2
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0answers
98 views

Normal cone to the union of sets

For a set $A \subseteq \Bbb R^n$ and a point $\bar{x} \in A,$ the limiting (Mordukhovich) normal cone of $A$ at $\bar{x}$ is defined as $$N(\bar{x},A):= \limsup_{x\to \bar{x}} \widehat{N}(x,A),$$ ...
2
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1answer
21 views

Is the bijection of $3$ functions valid when you take them as a 3d vector?

Let $F_1,F_2,F_3$ be three functions from $\mathbb R\to\mathbb R$. $F_1,F_2,F_3$ are bijective (we can say that they are strictly increasing on $\mathbb R$ by the bijection theorem they are bijective)....
2
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0answers
25 views

Reducing trace minimization to generalized eigenvalue problem

I'm looking for insight in solving the following optimization over symmetric matrices A and positive-definite H. $$R=\max_{A}\frac{\text{tr}(HA)^2+2\text{tr}(HAHA)}{\text{tr}(AHA)}$$ The paper ...
2
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0answers
114 views

How to compute some variations? From $\dot{f_\epsilon}=\dot{f}+\epsilon\dot{h}n+\epsilon h\dot{n}$ to $\delta\dot{f}=\dot{h}n+h\dot{n}$

I have a paper and on that paper I only can read: Let $f:\mathbb{S^{1}} \to \mathbb{R^2}$ be a function and $f_{\epsilon}=f+\epsilon hn$ where $\mathbb{S^1}$ is the unit circle and $\dot{f}^2=r^2$. ...
2
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0answers
17 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 ...
2
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0answers
172 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
104 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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0answers
68 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|\...
2
votes
1answer
61 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
92 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
votes
1answer
64 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
53 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}_M....