# 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.

164 questions
Filter by
Sorted by
Tagged with
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 $x, y \in \operatorname{dom} f$: ...
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 ...
156 views

69 views

259 views

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 ...
66 views

I think I am doing something wrong when combining Lagrange multiplier and Euler-Lagrange equation. I need to maximize a functional of the form: $$\int\!dx~{L(x, G, \dot{G})}~~~~~\text{where } L(a, b,... 1answer 109 views ### How to determine the path a particle that is bound to a vector field I've been trying to solve this problem but there are no resources that help. I've tried different approaches to solve this problem but every one of them leads to a dead end. I've found one approach ... 1answer 37 views ### Variational principal question regarding functions that have a minimum at the origin under a restriction. I'm going over some old assignments from a couple terms ago and have come across a problem from my variational principals module. I looked at the function in the hint and noticed that for some ... 1answer 41 views ### The explicit expression of \frac{Ī“F}{Ī“P} I'm writing simulation code of ferroelectric domain, and there is a math problem that I can't solve. The expression of F is$$ F = \frac{|\vec{k} \cdot \vec{P}(\vec{k})|^2}{k^2}. $$\vec{k} is a ... 2answers 691 views ### Permutation and combinations using chairs? After reading and watching a lot about permutation, combination and variation i still don't understand them fully. So i have two questions: How many ways are there to position 5 people on 10 chairs? ... 1answer 52 views ### Finding consumption function which maximizes utility I can across this question in my applied real analysis textbook that I'm having trouble with. It asks us to consider the utility function U(C) = \sqrt{e^{-rt}C}. I'm supposed to find the consumption ... 3answers 153 views ### Euler-Lagrange formula Let y:[-1,1]\to [2-1,2+1] be a C^1-smooth function, and F(y,y'):=y\sqrt{1+y'^2}. Suppose y(x) satisfy the Euler-Lagrange equation, i.e.$$\frac{\partial F}{\partial y}-\frac{d}{dx}\frac{\...
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 ...