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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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 ...
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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:...
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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 ...
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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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65 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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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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88 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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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 ...
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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}...
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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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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) = \...
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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 ...
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1answer
101 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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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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28 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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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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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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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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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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51 views

Error is the Projection on the Noise - Does the intuition carry over from least squares?

My question concerns a generalization of the following: Background It is a well known that the least squares problem (on some Hilbert space) \begin{align*} \|AX-Y \|_2^2 \to \min_{A}! \end{align*} ...
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1answer
54 views

How to simplify the Euler-Lagrange equation of Brachistochrone in this way?

I already know that in the Brachistochrone problem, we have Euler-Lagrange equation: $$\frac{1}{2y}\sqrt{\frac{1+y'^2}{y}}+\frac{d}{dx}[\frac{y'}{\sqrt{y(1+y'^2)}}]=0$$ To solve this equation, we ...
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20 views

About the equivalence of two definitions of topological linking

If $E$ is a Banach space, denote with $I$ the identity map of $E$, denote the space of continuous functions from $E$ to $E$ with $\operatorname{C}(E,E)$ and denote the space of homeomorphism from $E$ ...
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31 views

Minimizing a nonlinear functional over finite elements

I am looking for a reference (an url, a book, or a paper) that could help me in discretization and minimization of the following cost function $J\left(\omega\right)$, over 3D tetrahedrons (finite ...
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79 views

Variational formulation of a parabolic equation

I have the following problem $$ \dfrac{\partial u}{\partial t} - \Delta u + F(u)= f(x,t) $$ where $u \in L^2([0,T[;H^1(\mathbb{R}^n))$ with $\partial_t u \in L^2([0,T[;(H^1(\mathbb{R}^n))')$ I want ...
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12 views

Bounded-length path that maximizes work in non-conservative field

Given a "sufficiently well behaved", but not necessarily conservative, vector field ${\bf E}$ in $\mathbb{R}^3$ defined in a bounded domain $\Omega$, what is the maximum value of $$ \int_C {\bf E} \...
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80 views

Finding a specific set of functions. (Please, read the description)

The task is to find the sequence of functions $\lbrace f_n \rbrace _{n\geqslant1} \subset \mathbb{K}\mathbb{C}^{1} \left[ 0;1 \right]$ (where $\mathbb{K}\mathbb{C}^{1} \left[ 0;1 \right]$ is a set of ...
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22 views

Is the trapezoid rule integral operator $A$ a non singular matrix?

Let $A$ be the matrix of a linear bounded integral operator $A : L^2 \rightarrow L^2$, discretized by the trapezoid rule defined on the function space $BV([0, L])$, $L \in \mathbb{R}$. So, if $u \in ...
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67 views

Definition of fundamental eigenvalue and sign of fundamental eigenfunction.

Consider the following integral eigenvalue equation $u = \lambda Ku$, where $K\colon L^2(F)\longrightarrow L^2(F)$ is a symmetric, compact, self-adjoint operator with positive, continuous kernel of ...
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12 views

First variation of coordinate transformation

Let $\varphi: \mathbb{R}^3 \to \mathbb{R}^3$ be a smooth coordinate transform in $\mathbb{R}^3$. Also, let $\mathcal{E} = C(\mathbb{R}^3)$ to be a space of real, scalar-valued functions on $\mathbb{R}^...
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1answer
112 views

Existence and uniqueness of weak solutions to the homogeneous biharmonic equation.

I am currently trying to prove boundedness and coercivity of the bilinear operator $a \colon H^2_0(\Omega) \times H^2_0(\Omega) \to \mathbb{R}$ defined by $$ a(u, v) = \int_\Omega \Delta u \Delta v \,...
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73 views

Minimize Variational Function over Normalized Distributions

I am looking at finding literature to be able to minimize the following variational problem: Minimize, $$ \mathcal{F}\left[p(y|x)\right] = I(X;Y) + \beta \ \mathbb{E}_{p(x,y)}\left[ d(x,y) \right] $$...
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122 views

Anisotropic Poisson equation strong to weak form

I'm trying to arrive at the weak form from the following strong form for the Anisotropic Poisson problem: $$\begin{align*}-\nabla\cdot(\textbf{A}\nabla u) &= f \hspace{.5cm}\textrm{in }\Omega \\ ...
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1answer
109 views

Finding stationary point 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. ...
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24 views

Relation between two types of pseudomonotone function

As far as I know, there are two different definitions for pseudomonotone function. The first definition was introduced by H.Brezis in 1968: Pseudomonotone function in the sense of Brezis: Let ...
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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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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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35 views

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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23 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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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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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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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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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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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: ...