# 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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### Calculating the inverse of planar flows

I am trying to find a way to calculate the inverse of a planar flow. In general, I understand that with normalizing flows, one can simply go from one distribution to the other with the change of ...
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### Weakly lower semicontinuity on weighted Sobolev space

Let $W^{1, p}(\Omega; \mathbb{R})$ be the standard Sobolev space and $\Omega\subset\mathbb{R}^2$ a bounded Lipschitz domain. If $f\colon\Omega\times\mathbb{R}\times\mathbb{R}^2$ is a Caratheodory ...
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### Solution for $F = \int^{b}_{a}{f(y'(x)) dx} \mbox{ min }$ with $\frac{d }{dp}f(p) = 0$ for $f(x,z,p) = f(p)$ and $f$ is convex.

I tried to solve the following problem, but somehow my ideas seem to a dead end: For $D := \lbrace y \in C^1([a,b]; \mathbb{R}) |y(a) = y_1 , y(b) = y_2 \rbrace$ we define \begin{equation*} F(y) = \...
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### Can I move arbitrary function inside integral?

I have a term $\delta g\int_V dV$, where $\delta g$ is the variation of a continuous function g(x,y,z) and $V$ is an arbitrary volume of integration over some physical 3D body. Moreover, $\delta g=0$ ...
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### A transforms converts an unbounded sequence into bounded

Recently, I read a paper(here the link) about variational methods. It said that Then we study a maximizing sequence $\{(u_k,v_k)\}$ of the functional $I(u,v)$. Such a sequence may be unbounded. Using ...
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### First variation and Gâteaux derivatives

I have two questions about the relation between Gâteaux and functional derivatives. In calculus of variation textbooks, the first variation is usually defined as follows. Let $V$ be a 'function space' ...
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### Extremal of $J[y] = \int_{0}^{1}e^x \sqrt{1 + y'^2} dx$ where $y \in C^2[0,1]$

This is a problem from a mathematical contest. We have to find the form of the extremal for the variational problem $J[y] = \int_{0}^{1}e^x \sqrt{1 + y'^2} dx$ where $y \in C^2[0,1]$. The answer ...
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### Natural boundary conditions for a variational problem when $~f~$ is a function of higher order derivatives of $~y~$

I am not sure about the boundary conditions for a variational problem when $~f~$ is a function of higher order derivatives of $~y~.$ i.e. if we have: $~f(x,y,y',y'')~$ then what is the natural ...
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### Obstacle Problem Help

I am attempting to solve the chapter 1 problems in Avner Friedman's Variational Principles and Free Boundary Problems, for independent study. There are no solutions posted anywhere that I can find, ...
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### Doubt in derivation of Euler-Lagrange equations for image registration

I am reading this paper "On the Metrics and Euler-Lagrange Equations of Computational Anatomy" and it contains a derivation of the LDDMM equations for image matching. I am having trouble ...
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### Variational Calculus Optimization Problem with x-independent Lagrangian Solution via Beltrami Identity

Optimize $$\int_0^1 y^2 (y')^2 dx$$ subject to $y(0)=0, y(1)=1$ For x-independent Lagrangians it is easier to use Beltrami Identity: $$F-y'{\frac{\partial F}{\partial y'}}=C$$ I have used that ...
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### Existence of a plane curve with a given curvature $\kappa(s)$ that passes through two given points $P_1, P_2$ and arc length $L$.

As the title, says, is it possible to find a curve (i.e. prove its existence) whose curvature is equal to a given function $\kappa(t)$, begins in $P_1$, ends in $P_2$, and has arclength $L$? The ...
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### A homework question on compact embedding of Sobolev spaces

I have encountered the following issue on a homework question. There seems to be a gap in my knowledge and I cannot answer it. For the set up, let $\Omega \subset \mathbb{R}^{n}$ be a bounded domain ...
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### Variational derivative in time varying domain?

Consider a spatial domain $\Omega$ enclosed by a boundary $\partial \Omega$. Denote by $\zeta\in\Omega$ the spatial variable and by $x=x(\zeta,t)$ the state variable of the dynamical system decribed ...
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### Convert differential equation to variational statement

I know how to convert a differential equation with constant coefficients to variational form. But the question in the picture has non-constant coefficients. How do people get around that? Any ...