# Questions tagged [calculus-of-variations]

Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.

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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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### 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 ...
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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/...
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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 $S^2_0(M)$ ...
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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 ...
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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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### 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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### 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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### 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 ...
1 vote
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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 ...
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. ...