Questions tagged [calculus-of-variations]
This tag is for problems relating to the calculus of variations that deal with maximizing or minimizing functionals. This problem is a generalization of the problem of finding extrema of functions of several variables. In fact, these variables will themselves be functions and we will be finding extrema of “functions of functions” or functionals.
3,197 questions
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Why the functional $J[u] :=\int_{-1}^{1}x^2 u^{\prime 4} dx$ doesn't have the $C^1$ minimizer?
From Lecture Notes M6367; Variational Methods:
..."5. Weierstrass’ Example.
The problem of minimizing the following functional $J[u]
:=\int_{-1}^{1}x^2 u^{\prime 4} \mathrm{d}x$ with $u(−1) = −1$...
- 31
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Using P.S. condition to find another critical points
Let $X$ be a real Banach space. Let $f\in C^1(X,\mathbb{R})$ satisfy the Palais–Smale condition: any sequence $\{x_n\}$ such that $|f(x_n)|\leq C$, uniformly in $n$, while $f'(x_n)\rightarrow 0$ as $n\...
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Frechet/Gateaux derivative vs functional/variational derivative
I am trying to understand the concept of differentiability on Banach spaces. However, it seems that there is a distinction between concepts found in math books, namely Gateux and Frechet Derivatives ...
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Uniqueness of type of functional which is able to produce E-L equation in the form of $-\Delta u+D \varphi \cdot D u=f$.
I want to ask about the uniqueness of functional which is able to produce E-L equation in the form of $$-\Delta u+D \varphi \cdot D u=f.$$
The answer here said that the energy functional of $$
-\Delta ...
- 751
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Relation of singular values of restriction to the spectrum
I've not used singular before, so I hope this question is not silly or trivial.
I assume I have a finite nonempty real set $\mathbb{V}\subseteq \mathbb{R}$ and a potential function $V:\mathbb{Z}^2\to \...
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Constrained Lipschitz Minimum of a Variational Problem
Consider the following variational problem:
$$
E(u) = \int_{-1}^{+1} x^2 u'(x)^2, \hspace{2mm} u \in \{u \in \text{Lip}([-1,+1]) \, \vert \, u(-1)=-1, u(+1)=+1, \text{Lip}(u) \leq K\}, \, \text{for} \,...
- 1,352
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Stationary points - Gateaux vs Frechet derivative
In basic vector calculus one terms a point $f$ stationary for $E$ if $\nabla E(f) = 0$. On the other hand, in variational calculus we term $f$ stationary for $E$ if the first variations are zero at $f$...
- 2,438
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Differentiability of the Distance Function
PRELIMINARY
Let $\Omega \subseteq \mathbb{R}^n$ be an open set. We adopt the following notation:
\begin{align}
(x',x_n) &\equiv x \in \mathbb{R}^n \\
Q &\equiv \{(x',x_n) \,\vert\, |x'|<1, |...
- 1,352
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Proof of fundamental lemma of calculus of variation using DCT
Let $f \in C((a,b))$ and suppose that
$$ \int_a^b f(x) \phi(x) dx = 0 $$
for all $\phi \in C_c^\infty((a,b))$.
Then it follows that $f \equiv 0$.
$\textbf{Proof outline:}$
Choose a sequence $(\phi_k)\...
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How to compute functional derivative of Dirichlet energy with Dirac delta function?
Let $D\subset \mathbb{R}^n$ be a bounded domain.
I am interesting in computing the functional derivative of the functional
$$ E[\phi] = \int_D \delta(\phi) \lvert \nabla \phi\rvert^2\ \mathrm{d}x. $$
...
- 1,212
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Book recommendation : Variation of shape functionals on a surface
As someone with working knowledge of basics of surfaces, curvature, tensors, differential operators, I am looking for a good textbook which can help me learn calculus of variation on surfaces.
My main ...
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Linking Theorem vs Mountain pass theorem critical value
I'm considering the Linking theorem in Martin Schechter's book Linking methods in critical point theory.
Let $E$ be a Banach space, and let $\Phi$ be the set of mappings $\Gamma \in C(E \times [0,1], ...
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Obtaining coordinate Euler-Lagrange equations from coordinate free Euler-Lagrange equations
This is a follow-up to the recent question of mine.
Let $X$ be a $n$-dimensional configuration space. Consider a Lagrangian $L:TX\to\mathbb R$. A coordinate-free formulation of Euler-Lagrange ...
- 1,221
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A question about definitions of Bounded Variations for the one dimensional case.
I have a question regarding the definitions of functions of bounded variation.
I've been using the following definition, which I found in the book Variational Methods by Struwe:
$f \in BV(\Omega)$ if ...
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97
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Calculation in Dido's problem
In this book, I was stumped by a calculation. As picture below, the area enclosed by the closed curve is $A$. The length of the closed curve is $L$. Then, I have
$$
A=\frac{1}{2} \int_0^{2\pi} x^2 +y^...
- 6,956
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Approximation of Sub-Harmonic Functions by Continuous ones
Could someone point me to a reference of a proof of the following statement:
If $u$ is a subharmonic function (ie: upper semicontinuous and satisfying the sub-mean value theorem on spheres) on a ...
- 1,352
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Is it possible to express the least action principle as a Poisson bracket?
I am currently wondering whether it is possible to express the variation of the action in the least action principle as a Poisson bracket. To be more precise, let $q$ be a system of coordinates, $p$ ...
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Local Bounded Slope Condition
From now on, let $\emptyset \neq \Omega \subseteq \mathbb{R}^n$ be a bounded domain (open and connected) and let $g: \partial \Omega \to \mathbb{R}$.
Definition (Bounded Slope Condition)
We say that $...
- 1,352
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Understanding the meaning of terms in Euler-Lagrange equation
Consider a Lagrangian $L=L(x, v, t)$ which is a smooth convex function from $TX\times\mathbb E$ to $\mathbb R$, where $X$ is a configuration space.
The Euler-Lagrange equations read
$$\frac{\partial L}...
- 1,221
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Formal definition of a perturbation in variational calculus on manifolds
The following is an paraphrase of the statement of the Euler-Lagrange equations on Wikipedia.
Let $X$ be a smooth manifold. Let $\mathcal{P}(a,b,\mathbf{x}_a, \mathbf{x}_b)$ be the set of smooth paths ...
- 1,269
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Perturb the metric and estimate the distance
I'm reading comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces by John Lott. I have a question about the following passage.
In general, $d^{2}\left(p_{0}, ...
- 3,323
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Is there a text that approaches Calculus of Variations as a special case of results on Banach Spaces?
It seems to me (correct me if I'm wrong) that Calculus of Variation is a subset of Analysis in Banach Spaces (see this post for an example).
Is there any text that approaches more general results on ...
- 5,310
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Leibniz rule for $\delta$ and $d$ on the Diffeological space $\mathcal{F} \times M$ in Lagrangian Field Theory
In the context of Lagrangian Field Theory I am reading Quantum Fields and Strings: A Course for Mathematicians, Volume 1, Part 1, Classical Field Theory, chapter 2, wherein they denote $\mathcal{F}$ ...
- 570
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62
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Nehari Manifold for coupled Schrodinger equations
When I study this equation
\begin{align}
\begin{cases}
-\Delta u_{1}+\lambda _{1}u_{1}=\mu _{1}|u_{1}|^{2}u_{1}+\beta _{21}|u_{2}|^{2}u_{1}\ \ \ \text{in $\mathbb{R}^{n}$},
\\-\Delta u_{1}+\...
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Requirements for Euler Lagrange Equations
For the integral
$$I =\int_a^b F(y', y, x) \, \mathrm dx$$
I’ve seen the requiremts expressed for the Euler Lagrange equation expressed in 2 different ways, but I do not see how they are equivalent.
**...
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Interpretation of adjoint in optimal control problem
This is a follow-up question to my previous question.
I'm considering the following optimal control problem where the idea is to maximize the state $x$ using some control $u$, and we know the dynamics ...
- 10.1k
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Minimum functionality $\mathcal{J} = \int\limits_{x^2+y^2+z^2 < 1} (u_xu_y+u_xu_z+u_yu_z)dxdydz$
I am solving the following problem: Find the equation and boundary conditions for the function that provides the minimum to the functional. Additionally, I have a condition: $$u \bigg|_{x^2+y^2+z^2 = ...
- 389
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A Sobolev-type (?) inequality with an exponential term
Let $f$ be a smooth function over $[a, b]$ with $0<a<b$. Suppose $f(a)=f(b)=0$. Does there exist a constant $C>0$ such that the following inequality holds for all $f$?
$$\int_{a}^b 2xf'^2~\...
- 4,163
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Continuity of Lagrangian Multiplier function
I am considering a Calculus of Variation problem about minimizing $I(x)=\int_{0}^{1} \phi(x(t),x'(t)) dt$, where $x(t)$ is in the space of absolutely continuous function on $[0,1]$.
When the integrand ...
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Looking for references on Gateaux derivatives, especially for physics applications
I am interested in finding literature on Gateaux derivatives, particularly in the context of variational methods in physics. I believe that by reformulating physical variational principles using ...
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Unusual Optimal Control Problem - Question about interpretation of $\lambda$
I'm trying to understand what seems like a very basic optimal control problem, but I'm getting two solutions that appear to be different and I'd like some help with clearing up the discrepancy. The ...
- 10.1k
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Proof of equality case in Prékopa-Leindler inequality, properties of log-concave functions
I’m trying to understand the proof of equality case of Prékopa-Leindler inequality, as proved here (the statement of the inequality is (3.6), the proof of equality case is at pages 87-88).
What I can’...
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1
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Short proof of the Fundamental Lemma of Calculus of Variations
I want to prove the theorem in the general setting
Let $\Omega \subset \mathbb{R}^d$ be open and $\mathcal{D}(\Omega)$ be the space of compactly supported smooth functions in $\Omega$. If $f \in L^1_{...
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"Surface of Steepest Descent" Given Two Points
Given two points $O := (0, 0), P := (x, y)$ in $\mathbb{R}^2$ and assume a uniform downward (negative $y$-direction) gravitational field is applied. By considering all the possible curves and their ...
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Why this energy functional is not bounded from below
I am reading a book on variational method and they're talking about minimisation of energy functional under constraint. To give an example, the author talks about the following problem:
$$
\begin{...
- 4,546
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Variation formula for a smooth family of metrics
Let $E$ be a vector bundle over a Riemannian manifold $(M, g)$ endowed with inner product $(\cdot,\cdot )$ on the fibers. Denote by $\Gamma(E)$ the space of sections of the $E$. We define a metric on $...
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Confusion around Gateaux differential over functional space with boundary conditions
Consider the following definition of Gateaux differentiable:
For a function $$F: V\rightarrow \bar{\mathbb{R}}$$ where $V$ is a normed vector space, the directional derivative of $F$ at $u$ in the ...
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Variation forms with respect the metric
Let $\alpha$ and $\beta$ be differential $p$-forms on an $n$-dimensional manifold $(M,g)$. Consider the wedge product
$$
\alpha \,\wedge\, \star\beta
$$
where $\star$ is the Hodge star operator ...
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Taking derivatives w.r.t a function of integrals.
I am struggling to understand how the calculus of variations justifies the jump from one line to the next:
$$E[L] = \int \int \{f(\mathbf{x}) - t\}^2p(\mathbf{x},t)d\mathbf{x}dt$$
$$\implies$$
$$\frac{...
- 661
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Variation of sections of a vector bundle and infinitesimal symmetries of a functional
Let $M$ be an oriented Riemannian manifold and $F = \Gamma(E)$ the space of smooth sections of a vector bundle $E$ over $M$. We will equip $F$ with the structure of a smooth Fréchet manifold (see here,...
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3
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113
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Minimum energy function with average $0$ and defined boundary conditions
What continuous function $f$ solves the following minimization problem?
$$ \begin{array}{ll} \underset{f \in C([0,1])}{\text{minimize}} & \displaystyle\int\limits_0^1 f(x)^2 {\rm d}x \\ \text{...
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Derivative of the functional $ \frac{1}{4}\int\limits_{\Omega}|u|^{4} \, {\rm d}x $
Let $d > 1$ and let the set $\Omega \subset \mathbb{R}^{d}$ be open and bounded. Let the functional $J : W^{1,2}_{0}(\Omega,\mathbb{C}) \to \mathbb{R}$ be defined by
$$ J(u) := \frac{1}{4}\int_{\...
- 11
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0
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42
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Is there a simple, straightforward way to shift the Euler equation (and the function $F$) between coordinate systems?
I'm not currently a student; the material is being practiced purely for learning and independent research. I'm currently reading through Gelfand and Fomin's Calculus of Variations.
Note that Gelfand ...
- 1,881
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71
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What curve minimizes this integral when the values of $y$ are not specified at the end points?
This problem is adapted from Gelfand and Fomin's Calculus of Variations (Chapter 1, Problem 18). I'm not currently a student; the material is being practiced purely for learning and independent ...
- 1,881
1
vote
0
answers
85
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Find the general solution of the Euler equation corresponding to the functional $\omega(y) = \int \left(f(x) \sqrt{1 + y'(x)} \right)dx$
This problem is adapted from Gelfand and Fomin's Calculus of Variations (Chapter 1, Problem 18). I'm not currently a student; the material is being practiced purely for learning and independent ...
- 1,881
1
vote
0
answers
88
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$H^4$-regularity of minimizer
I am fairly new to the calculus of variations and have been studying a variational problem similar to a hinged plate on a convex polygonal domain.
Concretely I am studying the energy functional
$$
E(u)...
- 1,233
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0
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37
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Find the extremal for the functional $ \int \limits_a^b \left( \left(y(x) \right)^2 = \left(y'(x) \right)^2 + 2 y(x) e^{x} \right) dx $
This problem is adapted from Gelfand and Fomin's Calculus of Variations (Chapter 1, Problem 15.d). I'm not currently a student; the material is being practiced purely for learning and independent ...
- 1,881
2
votes
0
answers
51
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Find the extremal of the functional $\int \limits_a^b {\left(y' (x) \right)^2 \over x^3} \,dx$
This problem is adapted from Gelfand and Fomin's Calculus of Variations (Chapter 1, Problem 15.b). I'm not currently a student; the material is being practiced purely for learning and independent ...
- 1,881
1
vote
0
answers
29
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Variation of a functional that takes a vector function as an argument
I am trying to derive the expression given for the variation of the action, in the wiki article about Hamilton's principle : https://en.wikipedia.org/wiki/Hamilton%27s_principle, from a pure math ...
- 367
2
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0
answers
72
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Find the extremal of the functional $\int \limits_a^b \left((y (x))^2 + (y' (x))^2 - 2y \sin (x) \right)dx$
This problem is adapted from Gelfand and Fomin's Calculus of Variations (Chapter 1, Problem 15.a). I'm not currently a student; the material is being practiced purely for learning and independent ...
- 1,881