Questions tagged [calculus-of-variations]
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
515
questions
3
votes
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Thin-Plate-Spline understanding and solution
This is a migrated question from: Thin-Plate-Spline understanding and solution.
If I need to delete one of the questions let me know. I was suggested to post it here as well.
As I understand it a Thin-...
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-1
votes
2
answers
194
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Conditional expectation: commuting integration and supremum
Let $X$ and $A$ be compact Polish spaces endowed with Borel $\sigma$-algebras. Let $\mathcal{A} = X\times \mathcal{B}(A)$ be the $\sigma$-algebra consisting of cylinders whose projections on $A$ are ...
- 109
0
votes
0
answers
63
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What is the definition of a Brachistochrone curve in a non-Euclidean space?
I have a problem where I have to study "the geometric properties of the Brachistochrone curve in non-Euclidean spaces". But I am confused about the definition of the Brachistochrone Problem/...
- 1
2
votes
0
answers
33
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$1$-parameter family of metrics preserving the normal direction
Let $(M^n,g)$ be a compact Riemannian manifold with boundary, $n \geq 2$, and let $N$ be the unit outward normal to $\partial M$. Denote by $S^2(M)$ the symmetric covariant $2$-tensors, by $S^2_0(M)$ ...
- 2,001
2
votes
1
answer
123
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How to solve an optimization problem whose optimization variable is a function?
I would like to find an optimal probability density function (PDF) $f$. Given $b$,
$$ \begin{array}{ll} \underset {f} {\text{minimize}} & C \\ \text{subject to} & 1 + \frac{b}{x} \displaystyle\...
- 21
3
votes
0
answers
76
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A variant of the Laplace principle
$\newcommand{\R}{\mathbb R}\newcommand{\eps}{\varepsilon}$In $\R^d$ I am given a sequence of smooth functions $f_\eps(x)$ that converges uniformly to some $f(x)$, which is assumed to be a good rate ...
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1
vote
0
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71
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Nonlocal elliptic problem - what is its associated energy?
It is well known that for any smooth domain $\Omega\subset\mathbb{R}^N$ the energy functional (the one for which the Euler-Lagrange equation is our b.v.p.) associated to the following local problem:
$$...
- 1,154
0
votes
0
answers
59
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Euler-Lagrange equation with extra variable under integral. (Reference request)
Assume that we have a functional to optimize of the form $$\iiint L[x, f_y(y, z), f_z(y,z)]dxdydz.$$
As one may notice, the optimized function $f$ depends only on $y$ and $z$, which means that writing ...
- 467
5
votes
0
answers
83
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Varifold convergence of images of Sobolev maps
Suppose I have a sequence of maps $\{f_k:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}^{n+1}\}$ such that:
$f_k\rightharpoonup f_*$ weakly in $W^{1,p}(\Omega,\mathbb{R}^{n+1})$,
The images $\Sigma_k:...
- 151
5
votes
0
answers
90
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What normal variational vector fields are allowed for the area to be preserved?
Let $(M^3,g)$ be a Riemannian manifold with boundary and let $\varphi : \Sigma \to M$ be a two-sided proper embedding of a compact surface with boundary, with unit normal $N$.
If $\varphi$ is not ...
- 2,001
1
vote
2
answers
179
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How much "room" in inequality $\displaystyle \int_a^b \varphi' ov ~\mathrm{d}x \leq 0$
Let $[a, b]$ be a nonempty interval, $o \in C^1([a, b])$ be such that $o>0$ and $o'<0$ and assume we found some $v \in L^\infty(\mathbb{R})$ such that
\begin{equation}\tag{1}\label{1}
\int_a^b \...
- 115
1
vote
0
answers
64
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Reference request: theory for local minimizers in the calculus of variations
Let $F: \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ be the Lagrangian. We say that $f \in X$ is a local minimizer of the variational integral if for all compact sets $C \subset \...
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2
votes
0
answers
51
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Defining minimality 'through deformations'
Let $U \subset \mathbf{R}^{n+k}$ be a bounded open set, and $T \in \mathbf{I}_n(U)$ be an $n$-dimensional integral rectifiable current. Say that $T$ is stationary through (homological) deformations if ...
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0
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45
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A variational problem
I was studying the paper "Non-holonomic Euler-Poincaré equations and stability in Chaplygin's sphere" by D. Schneider (2002). A non-holonomic constraint - see equation (24) - is expressed as
...
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3
votes
1
answer
269
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Are all Helmholtz decompositions related?
Suppose $V\subset \mathbb{R}^3$ be non-empty and at least twice differentiable (Smooth) and let $S$ be the surface that encloses $V$ (for example a sphere). Let $\textbf{F}\in \mathbb{R}^3$ be a ...
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3
votes
1
answer
102
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Is this map (from the space of probability densities to the Wasserstein space) Lipschitz?
Let $p \in [1, \infty)$. Let $\mathcal P_p(\mathbb R^d)$ be the space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moments. Let $D_p$ be the collection of all Borel measurable ...
- 461
1
vote
1
answer
155
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Integral identity for critical points of the Ginzburg-Landau functional
I am reading a paper of Comte and Mironescu [CM96], where they discuss critical points $v = v_{\epsilon}: G \to \mathbf{C}$ of the (non-magnetic) Ginzburg–Landau functional
$E_\epsilon(v) = \frac{1}{2}...
- 4,524
0
votes
0
answers
40
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A maximization problem over a subspace of Sobolev space
Let $E\subset \mathbb R^2$ be a "nice" region (e.g. connected, open and bounded). Denote by $H^1_0$ the Sobolev space on $E$, i.e.
$$H^1_0:= \left\{f: E\to\mathbb R:\quad f \big|_{\partial E}...
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12
votes
0
answers
372
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A model of pillows
(The same system with slightly different questions has been asked in MSE.)
Let $\Omega\subset \mathbb{R}^2$ be some simply connected planar domain. We seek for a mapping $\mathbf{r}:\Omega\rightarrow \...
- 237
0
votes
0
answers
74
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Wasserstein compactness of sublevel sets of relative entropy
Let $\mathcal{P}({\mathbb{R}^d})$ denote the set of Borel probability measures on $\mathbb{R}^d$, and let $\pi \in \mathcal{P} (\mathbb{R}^d)$. It is known that the sets $\{ \mu \in \mathcal{P}(\...
- 568
2
votes
1
answer
119
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Gradient descent relaxation dynamics of a Euler-Lagrange equation
I want to minimize the functional
$$
F=\int{L(u)}dx,
$$
where $L= u_x^2-u^2$ is the Lagrangian function of the functional. Even if its Euler-Lagrange equation is easily found and solved, I want to try ...
- 159
4
votes
1
answer
143
views
Euler operator as part of a cochain complex
I am studying chapter 4 of Olver's "Applications of Lie groups to differential equations", about symmetries in differential equations coming from a variational principle.
The Euler operator ...
1
vote
1
answer
71
views
Strict convexity of the cost function is enough to ensure the existence and uniqueness of the optimal transport map
Let $X=Y = \mathbb R^d$ and $c:X \times Y \to [0, \infty)$ be Borel measurable. Let $\mu, \nu$ be Borel probability measures on $X,Y$ respectively. Let $\mathcal T$ be the set of all Borel measurable ...
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4
votes
1
answer
245
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Optimal Transport: how is this transport map Borel measurable?
I'm reading Theorem 1.17. and its proof at page 14 of Santambrogio's Optimal transport for applied mathematicians. The content is not hard but a little bit long (because of related detail). Please ...
- 713
2
votes
1
answer
146
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Hausdorff dimension of the non-differentiability set of a locally Lipschitz function
Let $f:\mathbb R^n \to \mathbb R$ and $E := \{x \in X : f \text{ not Fréchet differentiable at }x\}$. Then $E$ is Borel measurable. It is well-known that
Theorem If $f$ is convex, then the Hausdorff ...
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10
votes
1
answer
342
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Variation of the Einstein Hilbert action in a coordinate-free way
I am currently writing an expository paper on gauge theory and gravity and throughout the gauge theory part I have been able to mostly stay away from coordinates unless I wished to
provide specific ...
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1
vote
0
answers
72
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Gateaux differentiability
I have a question about the Gateaux differentiability of a map on a particular space. To be more specific, let $f: U \to X$ be a map where both $U$ and $X$ are Hilbert spaces.
Now, assume that there ...
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2
votes
0
answers
44
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A function $f_r$ where $f_r (x)$ is defined as the ratio between $f(x)$ and the average value of $f$ over $B(x, r)$
Let $E := \mathbb R^d$. Let $f:E \to \mathbb R_{>0}$ be continuous and integrable. For $r>0$, we define
$$
f_r (x) := \frac{f(x)}{ \frac{1}{|B(x, r)|} \int_{B(x, r)} f(y) \, \mathrm{d} y} \quad \...
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6
votes
1
answer
251
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What happens if we consider functions of bounded variation that are not in $L^1$?
A function $f \in L^1(\mathbb R^n)$ is said to be of bounded variation if there exists a constant $C \geq 0$ such that
$$
\int_{\mathbb R^n} f(x) \operatorname{div} \phi(x) \; dx
\leq
C \sup_{ x \in \...
- 4,516
4
votes
2
answers
229
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Discrete isoperimetric problems
It is well-known that among all planar curves, the circle — invariant under $O(2)$ — has the best isoperimetric ratio. Similarly, among all $n$-gons, the regular $n$-gon — invariant under the dihedral ...
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1
vote
1
answer
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Optimal transport: the existence of an optimal pair of $c$-conjugate functions
$\newcommand{\diff}{ \, \mathrm d}$
Let
$X,Y$ be Polish spaces,
$\mathcal C_b(X)$ the space of all real-valued bounded continuous functions on $X$,
$\mathcal P(X)$ the space of Borel probability ...
- 713
2
votes
0
answers
113
views
Optimal transport: how is the use of disintegration theorem valid in this construction of $\widetilde{\phi}$?
Let $X,Y$ be Polish spaces and $\mathcal P(X)$ the space of all Borel probability measures on $X$.
Fix $\mu\in \mathcal P(X), \nu \in \mathcal P(Y)$. Let $\pi \in \Pi(\mu, \nu)$, i.e., $\pi \in \...
- 713
1
vote
1
answer
111
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Optimal transport: how $\varphi^c$ can be written as $\varphi^c = \lim _{\ell \rightarrow \infty} \psi_{\ell}$?
Let $X,Y$ be Polish spaces and $c:X \times Y \to [0, \infty]$ lower semi-continuous. There is a sequence $(c_\ell)_{\ell \in \mathbb N}$ with $c_\ell:X \times Y \to [0, \infty)$ of bounded Lipschitz ...
- 713
1
vote
0
answers
43
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$c$-cyclical monotonicity: does this proof hold if $f \equiv +\infty$ or $\int c \mathrm d \gamma = +\infty$?
I'm reading the proof of Theorem 1.38. from section 1.6.2 $c$-cyclical monotonicity and duality of Santambrogio's Optimal transport for applied mathematicians.
My understanding: It seems for the ...
- 713
1
vote
0
answers
23
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Convergence of a sequence of minima of a family of optimal control problems
Let $(\lambda_k)$ be an unbounded and increasing sequence of real numbers, and $f:\mathbb{R}^n \ \times \mathbb{R}^m \rightarrow \mathbb{R}^n$ be sufficiently smooth such that $\dot{x}(t) = f(x(t),u(t)...
- 1,532
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vote
0
answers
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elaboration on the equation of directional derivative that lead to steepest gradient descent [closed]
I am reading the book
Ian Goodfellow and Yoshua Bengio and Aaron Courville, Deep Learning, MIT Press, 2016.
I am reaching to the point about directional derivative. Given the $u$ as the unit vector ...
- 11
0
votes
0
answers
101
views
Concentration compactness lemma and the best Sobolev constant
It is well known that the best Sobolev constant can be achieved on $\mathbf{R}^n$. More precisely, we have the following theorem (A):
Let $\frac{1}{q}=\frac{1}{2}-\frac{1}{n}$, $$S=\inf\limits_{{u\in ...
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1
vote
0
answers
32
views
Fibrewise coordinates in a neighborhood of a graph of a continuous curve
Let $M$ be a smooth manifold, $\dim M=n$ and $\gamma:[0;1]\to M$ be continuous. Is it true that there exists local coordinates $(y^0,\ldots,y^n)$ in a neighborhood $V$ of the graph $\{(t,\gamma(t)),t\...
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3
votes
1
answer
88
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Equivalent definition of the Kantorovich-Fisher-Rao distance
I am reading this paper
"A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows"
(https://arxiv.org/abs/1602.04457)
and in the proof of Proposition 2.2, basically, if the measure ...
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6
votes
1
answer
333
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What is the current status on bad tangent cones at isolated singularities?
Let $M^8 \subset B^9 \setminus \{ 0 \} \subset \mathbf{R}^9$ be a properly embedded, stable minimal hypersurface. Suppose that $0 \in \overline{M}$ is an isolated singularity of the surface.
Question.
...
- 4,524
3
votes
1
answer
196
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Tangent cones at zero and infinity to minimal surfaces
Let $n \geq 2$, and let $M^n \subset \mathbf{R}^{n+1}$ be a minimal surface with $0 \in M$ and finite ($n$-dimensional) area growth:
$\operatorname{limsup}_{R \to \infty} R^{-n} \lVert M \cap B_R \...
- 4,524
1
vote
0
answers
92
views
Is it possible for the Lagrange multiplier to satisfy some constraints themselves?
I am using the field-theoretic langauge, so that we think of some action functional
\begin{equation}
S[f_l,T_{ij}]=\int_0^1 dt \int_{[0,1]^3}d^3\overrightarrow{x}\mathcal{L}(f_l(\overrightarrow{x}, t),...
- 1,791
1
vote
0
answers
58
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In the context of field-theoretic classical Lagrangian mechanics, can we choose the Lagrange multipliers to be time-independent? - from Physics SE
I originally posted this question on Physics SE, but I think it is more like a math question since I need rigorous justification.
Could anyone please provide any insight to the below question:
Let us ...
- 1,791
4
votes
0
answers
84
views
Linking theorem
In 1978 Rabinowitz obtained the classical "Linking theorem", which is used to solve, for example the classical problem:
$$
\begin{cases}
-\Delta u = \lambda u + |u|^{p-2}u, \Omega \\
u = 0, \...
5
votes
0
answers
117
views
Functional inverse problem based on a variational principle
I am trying to solve an inverse problem based on variational principle.
I will first present a forward problem that is already solved, and then present the inverse problem that I am trying currently ...
1
vote
0
answers
75
views
Variational problem with constraint
Let $D\subseteq [0,2\pi]\times [0,2\pi]$ and ${D}^\complement$ be the complementary region, i.e.
$D \cup {D}^\complement = [0,2\pi]\times [0,2\pi]$ and
$D\cap {D}^\complement = \emptyset$.
I would ...
5
votes
0
answers
166
views
Equality of weak solutions for inner products inducing equivalent norms
This is a repost of a now-deleted MSE question that did not get any comments or answers.
$\textbf{Background}$: This question is mainly about two basic questions: How do we systematically obtain the ...
- 101
2
votes
1
answer
189
views
Optimal transport for applied mathematicians: how does $\varphi (x) = \inf_{y \in Y} [c(x, y) - \psi (y)] \neq -\infty$ follow in Theorem 1.37?
I'm reading a proof of Theorem 1.37 from Santambrogio's Optimal transport for applied mathematicians: calculus of variations, PDEs, and modeling. First, I quote related definitions. Let $X,Y$ be ...
- 461
1
vote
1
answer
139
views
How do I integrate this inequality that appears in a paper of Rabinowitz?
Sorry if this is too easy for MO, but I found it in a research paper, so I thought that it was worth posting here.
I was reading a paper by Rabinowitz(this one to be more precise) and I came across ...
- 113
1
vote
1
answer
132
views
Let $f$ be convex and $A$ a Borel subset of $\mathbb R^d$ on which $f$ is differentiable. Is the gradient $\nabla f: A \to \mathbb R^d$ measurable?
Let $X := \mathbb R^d$, $\lambda^d$ be the $d$-dimensional Lebesgue measure on $X$, and $f:X \to \mathbb R$ convex. Then there is a Borel set $N \subset X$ such that $\lambda^d (N) = 0$ and $f$ is ...
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