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# 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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### 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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### 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 ...
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### 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. ...
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### 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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### 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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### 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/...
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\... 2answers 95 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$? Does ... 1answer 53 views ### Why minimize$ \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)$with ... 1answer 55 views ### An example for a stable harmonic map which is not a local minimizer I am looking for an example of a harmonic mapping between compact Riemannian manifolds, which is stable but not a local minimizer of the Dirichlet energy. A harmonic map$f:M \to N$is said to be ... 1answer 242 views ###$\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: $$\hat{N}_{\varepsilon }\left( \bar{x};\Omega \right)... 1answer 88 views ### How to determine the path a particle that is bound to a vector field I've been trying to solve this problem but there are no resources that help. I've tried different approaches to solve this problem but every one of them leads to a dead end. I've found one approach ... 1answer 32 views ### Variational principal question regarding functions that have a minimum at the origin under a restriction. I'm going over some old assignments from a couple terms ago and have come across a problem from my variational principals module. I looked at the function in the hint and noticed that for some ... 1answer 41 views ### The explicit expression of \frac{δF}{δP} I'm writing simulation code of ferroelectric domain, and there is a math problem that I can't solve. The expression of F is$$ F = \frac{|\vec{k} \cdot \vec{P}(\vec{k})|^2}{k^2}. $$\vec{k} is a ... 2answers 392 views ### Permutation and combinations using chairs? After reading and watching a lot about permutation, combination and variation i still don't understand them fully. So i have two questions: How many ways are there to position 5 people on 10 chairs? ... 1answer 48 views ### Finding consumption function which maximizes utility I can across this question in my applied real analysis textbook that I'm having trouble with. It asks us to consider the utility function U(C) = \sqrt{e^{-rt}C}. I'm supposed to find the consumption ... 3answers 138 views ### Euler-Lagrange formula Let y:[-1,1]\to [2-1,2+1] be a C^1-smooth function, and F(y,y'):=y\sqrt{1+y'^2}. Suppose y(x) satisfy the Euler-Lagrange equation, i.e.$$\frac{\partial F}{\partial y}-\frac{d}{dx}\frac{\... 0answers 107 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$\dot{f}^2=r^2$. ... 0answers 62 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 $$\... 0answers 93 views ### Optimization in Banach space: Find functions that minimize the supremum of a convex operator. Let D \subset \mathbb{R}^n be compact. Denote by C(D, \mathbb{R}^n) the space of continuous functions from D to \mathbb{R}^n. Let K be a real, symmetric, positive-definite n \times n ... 1answer 58 views ### Simple exercise with differentials Given the diferentials$$ $$d'Q=K(x-x')d'x$$  $$d'Q'=K(x'-x)d'x'$$ $$where K is a constant, I need to show that$$ \... 0answers 78 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 ... 1answer 51 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: $$L[q_1,...,q_m] -... 0answers 50 views ### Is there always a non-trivial homotopy class of maps which minimizes the Dirichlet energy? Let M,N be d-dimensional oriented Riemannian manifolds, possibly with boundary, M compact. Let E:C^{\infty}(M,N) \to \mathbb{R} be the Dirichlet energy, i.e.$$ E(f)=\int_M |df|^2 \text{Vol}... 0answers 89 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 ... 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 $$\frac{d\theta}{dx}\Bigg{|}_{x=0} = 0$$ $$\theta(1) = \... 0answers 36 views ### variation method Define F[\phi]=\int dr \{V(\phi)+\frac{1}{2} \kappa \vert \nabla \phi \vert^2 \}, then how can we deduce that \frac{\delta F}{\delta \phi}= \frac{d V}{d \phi}-\kappa \nabla^2 \phi ? I feel ... 1answer 115 views ### The integral under the variation \delta sign In physics books on classical field theory, the authors usually define the action as$$ S = \int\mathcal{L(\phi,\partial_\mu\phi)d^4x} $$where \mathcal{L} is the lagrangian density. Then, ... 1answer 140 views ### what is a variational form in optimization? This is probably a pretty basic question, but I can't figure out what people mean when they say "variational forms" in optimization. For example, in this paper I'm reading, the variational form of a ... 1answer 841 views ### What are prerequisites to Terry Tao's An Introduction To Measure Theory? I am an economics student and want to study mathematics, variational analysis in particular, with measure approach but since I am ignorant of measure theory I decided to try this book but I still find ... 1answer 50 views ### How to take derivative of integral of function? I'm reading a textbook where it forms a Lagrangian function$$ L = \int_0^1 f(x)^{1 - \frac{1}{\alpha}}dx - \lambda\int_0^1 g(x) f(x) dx$$But how do you take the derivative of this thing? The ... 2answers 68 views ### Using Hamilton's principle to derive Newton's equations of motion in parabolic coordinates I have recieved a very hard (optional) assignment on variational calculus, and I have not got a clue where to start other then stating the Euler-Lagrange equations. Here is the problem: According to ... 1answer 79 views ### With the solution of the Euler-Lagrange equation prove the following equation Let \Omega \subset \mathbb{R}^n and F=F(z,p):\Omega \times \mathbb{R} \times \mathbb{R}^n be smooth and independent of x\in\Omega. Let u be a solution of the Euler-Lagrange equation of \... 1answer 77 views ### Is the normal cone well defined? I am reading Mordukhovich's Variational Analysis and Generalized Differentiation, and the first definition there is that of a \epsilon-normals Given a Banach space X, and \Omega\subseteq X, ... 1answer 630 views ### Solving PDEs using the Ritz method on variational calculus problem (Student questions) I'm reading the book "Conduction Heat Transfer" by Vedat S. Arpaci. I'm currently at chapter 8 (I didn't read the rest of the book, though), which talks about the Variational Formulation - Solution by ... 1answer 36 views ### Poincaré inequality for Lipschitz functions with bounded domain Let u\in W^{1,\infty}(B_h(0),\mathbb R^n), where B_h(0)=\{x\in\mathbb R^n:|x|<h\}. From the Poincaré inequality we know that$$ \|u-\mathrm{Id}-\frac{1}{\mathrm{Vol}(B_h(0))}\int_{B_h(0)}(u-\... 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)...
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 ...