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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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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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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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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0answers
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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0answers
70 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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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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0answers
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/...
3
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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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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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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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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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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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174 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
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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
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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....
2
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0answers
111 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 ...
2
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0answers
36 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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0answers
62 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 $V$ be ...
2
votes
1answer
148 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, ...
1
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16 views

Moser Iteration for Laplacian with Hardy potential

I am reading the following proof of Cao-Yan's 2010 CVPDE Paper. It's a property for solutions of Laplace equation with Hardy potential $|x|^{-2}.$ The space dimension $N \geq 3.$ Their proof is in the ...
1
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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 ...
1
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0answers
19 views

Approximating KL divergence between mixture of gaussian and standard normal

I am reading this paper, and I don't get the proposition given in the appendix of this paper. Proposition 1 Let $K,L \in \mathbb{N}$, a propbability vector $(p_1,...,p_L)$, and $\Sigma_i\in\...
1
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0answers
39 views

How to prove this identity for inner products?

$\newcommand\normt[1]{\left\lVert#1\right\rVert_{L^2}}$ We want to minimize the distance function or its square, $$f(\zeta_1,\zeta_2) = \normt{u(x)-\sum_{k=1,2}\sigma_kQ(x-\zeta_k)}^2 = \int \left[u(...
1
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0answers
61 views

Generalized definition of the Green Formula (In Sobolev Spaces)

Given the classical version of Green's Formula: Let $\Omega$ be an open bounded subset of $\mathbb{R}^{n}$ and $u, v \in C^{2}(\overline{\Omega})$. \begin{equation} \int_{\Omega} Dv \cdot Du \ dx = -...
1
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1answer
97 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 ...
1
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0answers
42 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 ...
1
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0answers
25 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^{-1}...
1
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0answers
77 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: $f$ ...
1
vote
1answer
99 views

How is the Lagrangian function homogeneous in the velocities?

I'm working through The Variational Principles of Mechanics by Cornelius Lanczos. In chapter 6, section 10, he says that the Lagrangian function $$L_1 = L\left(q_1,\dotsc,q_{n+1}; \frac{q_1'}{q'_{n+1}}...
1
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0answers
53 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*} ...
1
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1answer
91 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 ...
1
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0answers
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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0answers
49 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 ...
1
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0answers
60 views

Two-way anova or One-way anova

I have to tested multiple reinforcement learning systems and record their performances [test-accuracy] based on two categories of [system-type] and [communication-level]. The system-type can be either ...
1
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0answers
98 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 ...
1
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0answers
14 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} \...
1
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0answers
81 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 ...
1
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0answers
23 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 ...
1
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0answers
101 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 ...
1
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0answers
16 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}^...
1
vote
1answer
177 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 \,...
1
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0answers
636 views

Is Euler-Lagrange equation necessary and sufficient for minimization in a variational problem?

To the best of my knowledge, the function that minimizes of the integral posed in calculus of variations must also satisfy the Euler-Lagrange equations. In other words, the Euler equations are a ...
1
vote
0answers
77 views

Minimize Variational Function over Normalized Distributions

I am looking at finding literature to be able to minimize the following variational problem: Minimize, $$ \mathcal{F}\left[p(y|x)\right] = I(X;Y) + \beta \ \mathbb{E}_{p(x,y)}\left[ d(x,y) \right] $$...
1
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0answers
161 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 \\ ...
1
vote
1answer
125 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. ...