Questions tagged [calculus-of-variations]

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

Filter by
Sorted by
Tagged with
2 votes
0 answers
80 views

How can I show that the exponents are not blowing up?

I have this situation: supposing that $\Omega$, is an open, bounded subset of $\mathbb{R}^N$, $N>2$. Given $t_0>0$ and $\overline{n}>0$, let's consider $\overline{t}=t_0+1$, $m>\overline{t}...
user avatar
4 votes
1 answer
154 views

Uniqueness of Kantorovich potentials?

$\newcommand{\R}{\mathbb R}$Take $\Omega\subset \R^d$ bounded, convex, and smooth. Consider the optimal transport problem with cost $c(x,y)=\lvert x-y\rvert^2$, leading to the quadratic Wassersein ...
user avatar
6 votes
1 answer
350 views

Is there any hope to prove that $g(x)>-4$ if $f(x)<0$?

I have these two functions for $x>0$, $\beta>0$ and $\alpha$ (all reals) $$ f(x)= \frac{\alpha \; \sin (\beta \; x)}x+4 \cos (\beta\; x) ,\qquad\qquad\qquad\qquad\qquad\qquad\\ g(x)=\frac{\...
user avatar
  • 61
2 votes
1 answer
128 views

$2\mathrm{d}$ area maximizing short embeddings

Think of a beach ball on an pool of water or sand. Let $\left(\mathcal{M}^2,g\right)$ be a surface homeomorphic to a sphere, endowed with a Riemannian metric $g$, and $\left(\mathcal{N}^2,h\right)$ a ...
user avatar
1 vote
0 answers
92 views

What is the role of of continuity in this proof of Kantorovich duality?

I'm reading the proof of Kantorovich duality from Villani's book Topics in Optimal Transportation. Let $X$ and $Y$ be Polish spaces. Let $P(X), P(Y)$ be the spaces of all Borel probability measures ...
user avatar
  • 135
0 votes
0 answers
77 views

Has the mixture of forward and backward finite difference existed?

Given a function $ f(x) $, there are forward and backward finite differencs, whose definitions are given in the following. By forward one, we mean $ \Delta f(x) = f(x+d)- f(x) $, $(d>0)$; and by ...
user avatar
3 votes
0 answers
140 views

$C^1$-regularity of solution of a Dirichlet problem

I've been stuck trying to hunt down a proof/reference for a certain regularity result which I need for my thesis (all the authors I've consulted refer to it as "well-known"). Consider the ...
user avatar
3 votes
0 answers
57 views

Prescribing variations that preserve the area

Let $(M^3,g)$ be a Riemannian manifold and let $\varphi : \Sigma \to M$ be a two-sided embedding of a closed surface into $M$, with a unit normal $N$. Suppose that $\varphi$ is a regular point of the ...
user avatar
3 votes
1 answer
127 views

Area of a deformation of a closed surface

Let $(M^3,g)$ be a complete Riemannian manifold. Fix a two-sided immersion $\varphi : \Sigma^2 \to M$ from a closed surface into $M$, with unit normal $N$. Given $f \in C^\infty(\Sigma)$ and $\alpha : ...
user avatar
4 votes
0 answers
97 views

Density of smooth function in the calculus of variations

In the non-convex calculus of variations, in the context of non-linear elasticity, the following classes of mappings $u:\Omega\to\mathbb{R}^n$, $\Omega\subset\mathbb{R}^n$, were introduced by John ...
user avatar
6 votes
1 answer
251 views

An $L^1$ function but (really) no better?

Question: For a smooth, bounded domain $\Omega\subset \mathbb R^d$, does there exist a function $u\in L^1(\Omega)$ such that $u\not\in L^\Phi(\Omega)$ for any Orlicz space $\Phi$? For the definition ...
user avatar
2 votes
1 answer
143 views

Sieve theory through variational principles

Disclaimer: I'm just starting to read Sieve Methods by Halberstam and Richert, so my present knowledge of the subject is close to zero, but it made me wonder if some connection to physics could exist, ...
user avatar
5 votes
1 answer
144 views

Is every set with finite $\mathcal{H}^{n-1}$ measure a set of locally finite perimeter?

Given a measurable set $E \subset \mathbb{R}^d$, with $\mathcal{H}^{d-1} (\partial E) < +\infty$, is it true in general that $E$ is a set of locally finite perimeter? that is, is it true that $\...
user avatar
  • 623
6 votes
1 answer
241 views

Lipschitz property of the symmetric rearrangement

I'm currently reading Talenti's paper "Best constant in Sobolev inequality" and am rather stuck on an argument on pg 363 (or pg 11 if you're reading the pdf). In this section of the paper, ...
user avatar
  • 163
0 votes
1 answer
71 views

A question about Marino–Prodi perturbation

In this paper N. Ghoussoub, the author claims the following version of Marino–Prodi perturbation, that is : Let $H$ a Hilbert space. Let $f\in C^2(H, \mathbb{R}),$ $K$ is a compact subset of $K_c$ (...
user avatar
-1 votes
1 answer
122 views

What functions are equal to their symmetric decreasing rearrangement?

I am trying to understand the set $$\mathcal{A}=\{f:f\text{ non-negative, measurable}, \text{vanishes at infinity and } f(x)=f^*(x)\}$$ where $f:\mathbb{R}^n\to \mathbb{R}$ is vanishing at infinity if ...
user avatar
  • 443
4 votes
0 answers
220 views

Malliavin calculus and geometric interpretation of $\nabla \cdot ({\nabla F(x)}{\|\nabla F(x)\|^{-2}})$, with regards to the surface $S = \{F = 0\}$

Let $F:\mathbb R^n \to \mathbb R$ be a "sufficiently regular" function. For any $k \ge 1$ and $x \in \mathbb R^n$, define $$ \alpha_k(x) := \nabla \cdot \left(\dfrac{\nabla F(x)}{\|\nabla F(...
user avatar
  • 5,728
3 votes
0 answers
55 views

Existence of ground state solutions for the critical exponent

I have been recently reading Kwong's paper on the uniqueness of positive solutions for the equation $\Delta u-u+u^p=0$ in $\mathbb{R}^n$. The authors show that the above equation has a unique positive ...
user avatar
  • 443
0 votes
0 answers
37 views

Minimal condition on set for an optimisation problem

We fix $\Omega \subset \mathbb{R}^{2}$ an open set. My question is: what are the minimum conditions we need on $E \subset \Omega$ such that the following optimisation problem: $$ \sup\{ \int_{E}(\...
user avatar
  • 181
1 vote
1 answer
121 views

Derivatives of infimum in variational problem

Define $$ F(\lambda,x):=g(x)+\lambda\int\limits_{\partial_e Y}f(y)d\mu_x(y) $$ where $Y$ is a convex space and $w^*$-compact, moreover $Y$ forms a Bauer simplex (in particular $Y$ corresponds to some ...
user avatar
  • 53
1 vote
0 answers
79 views

Minimizer for a certain variational problem

In the process of estimating certain hitting probabilities for Brownian motion I've run into the following variational problem. Let $\mathbb{D}\subseteq{\mathbb{C}}$ be the unit disk in the plane, ...
user avatar
1 vote
1 answer
52 views

Does a minimiser exist for this Gaussian-like functional?

Let $f(x)$ be a strictly increasing function such that $\lim\limits_{x\to\pm\infty}f(x)=\pm\infty$ and $\lim\limits_{x\to+\infty}f'(x)e^{f(x)-x}=+\infty$. If $f(a)=0$ for some prescribed $a\in\Bbb R$, ...
user avatar
4 votes
1 answer
196 views

Is $\int_{-c}^c |A \cap (x + A)|\, dx$ maximized when the measurable subset $A \subseteq \mathbb R$ is an interval centered at the origin?

Let $A$ be a nonempty measurable subset of $\mathbb R$, with Lebesgue measure $|A|=1$, and let $c>0$. Define the scalar $I(A)$ by $$ I(A) := \int_{-c}^c |A \cap (x + A)|\, dx, $$ where $x+A := \{x +...
user avatar
  • 5,728
2 votes
0 answers
65 views

Covariant momenta associated to higher order Lagrangians

Let $\pi:Y\rightarrow X$ be a fibered manifold with fibered coordinates $(U,x^i,y^\rho)$ (whenever local calculations are needed) and $m$ dimensional base $X$ ($\dim X=m$). Suppose that $L\in\Omega^m_{...
user avatar
0 votes
0 answers
23 views

References for discrete calculus of variations

What would be good references for discrete calculus of variations? For applications such as minimizing a functional not on a $[0,1]\times[0,1]\to \mathbb{R}$ but on a bitmap image that approximates ...
user avatar
  • 1,942
4 votes
1 answer
234 views

Which set of functions admits the existence of the minimizer?

Let $a,b \in \mathbb R$ and consider the functional $J$ on $X$: $$J[u] = \int_0^1 \left( (u'(x))^2 -a)^2 + b \ln (1+ u^2(x))\right) dx$$ Providing reasons specify if the $\inf J$ over $X$ is attained ...
user avatar
9 votes
0 answers
271 views

Is there a variational interpretation for the equation $\operatorname{div}(\star \circ \bigwedge^k df\circ \star )=0$?

$\newcommand{\id}{\operatorname{Id}} \newcommand{\R}{\mathbb{R}} \newcommand{\TM}{\operatorname{TM}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Cof}{\operatorname{Cof}} \newcommand{\Det}{\...
user avatar
  • 6,356
3 votes
3 answers
160 views

Non convex optimization problem in $W_0^{1,2}$

Let $0< \alpha \ll 1$. I'm trying to minimize $\int_0^\pi |f'|^2 dx$ over the functions $f \in W_0^{1,2}([0,\pi])$ (or at least find "good" lower bound in terms of $\alpha$) such that ...
user avatar
  • 331
3 votes
1 answer
98 views

The regularity theorem, a non-regular minimizer problem

During my self study to the calculus of variations I come across this problem. Because of my search, I know what I wanted to do but I need some help to do them. The function $f:[-1,1] \times \mathbb R ...
user avatar
1 vote
1 answer
56 views

What is the maximum possible coefficient of variation for data taking values within a specified range?

I have a question that seems very basic, and yet I have not managed to find an answer after probably several hours of Google-searching. Fix $0<a<b<\infty$, and let $\mathcal{P}_{[a,b]}$ be ...
user avatar
2 votes
0 answers
57 views

Best constant for Sobolev-type inequality

I am currently reading a paper from Del Pino/Dolbeault about optimal constants of GNS inequalities (http://capde.cmm.uchile.cl/files/2015/06/pino2002.pdf). The author wants to prove the following GNS ...
user avatar
3 votes
1 answer
222 views

Special version of Tonelli’s theorem

I am trying to prove this theorem. I have not found anything similar to it on the internet. Special version of Tonelli’s theorem Assume that the functions $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{...
user avatar
5 votes
1 answer
178 views

On the fundamental solution for elliptic PDE

In the well known paper by Littman-Weinberger-Stampacchia "Regular points for elliptic equations with discontinuous coefficients", the authors were able to prove the validity the following ...
user avatar
  • 143
1 vote
1 answer
120 views

Definition of Euler-Lagrange equation and properties, where can I find?

I'm studying a paper and in the introduction appears the following: It is well known that existence of critical points and solvability of Euler-Lagrange equations are related, and there is and ...
user avatar
3 votes
1 answer
139 views

Is the ball ratio theorem for Radon–Nikodým derivative known for general metric spaces?

Given two non-negative Borel measures $\mu$, $\nu$ on $\mathbb{R}^n$, that are finite on compact sets, such that $\nu\ll\mu$, it is well known that $$\frac{d\nu}{d\mu}(x)= \lim_{\epsilon\to 0} \frac{\...
user avatar
  • 173
2 votes
0 answers
98 views

Convergence of Gibbs distribution to Dirac measure [closed]

Consider the probability density function on $R^d$ for a continuous function $F: R^d \to R$: $$ q_{\varepsilon}(x) = \frac{1}{Z} \exp\left(-\frac{1}{\varepsilon} F(x)\right). $$ Denote $x^* = \arg \...
user avatar
4 votes
1 answer
179 views

On some convergence theorems by Felix E. Browder (1967)

I have been reading Felix E. Browder's Convergence Theorems for Sequence of Nonlinear Operators in Banach Space and I was hoping I could find answers to a couple of questions I have about the paper. ...
user avatar
0 votes
0 answers
66 views

Calculus of variations for fractions of integrals

I would like to know if optimisation problems of the following form have already been studied: Let $\mathcal{F}$ be a class of functions $f:\mathbb{R} \rightarrow \mathbb{R}$. Minimize the following ...
user avatar
2 votes
0 answers
100 views

Hilbert's fourth problem and a non-linear integral transform

The following nonlinear integral transform takes continuous functions defined on the cylinder $\mathbb{R} \times S^1$ to $C^2$ functions defined on the plane $\mathbb{R}^2$: $$ \mathcal{A}f (x,y) := \...
user avatar
  • 12.9k
6 votes
1 answer
302 views

Which geometric variational problems admit an entropy identity?

Question. My question in broad terms: when and how can entropy be used in geometric variational contexts to study sequences of critical points, such as the two results described below? [See at the ...
user avatar
  • 2,462
7 votes
2 answers
444 views

A little problem in PDE or function analysis

Let $E$ be the usual sobolev space $H^{1}_{0}(\Omega)$ on a smoothly bounded domain $\Omega$, $E_{k}$ be its subspace spanned by the first $k$ eigenfunctions of the Laplace operator, i.e. $$E_{k}:=\...
user avatar
2 votes
4 answers
215 views

EM-wave equation in matter from Lagrangian

Note I am not sure if this post is of relevance for this platform, but I already asked the question in Physics Stack Exchange and in Mathematics Stack Exchange without success. Setup Let's suppose a ...
user avatar
  • 61
2 votes
1 answer
164 views

Bounded variation of the partial derivatives of a convex function

Let $f:\mathbb R^2\to\mathbb R$ be a convex function. For simplicity, assume that $f\in C^1$. A general theorem which can be found in the book of Evans and Gariepy says that the gradient $\nabla f$ is ...
user avatar
3 votes
0 answers
86 views

When is the least-area surface unique?

Let $M^{n-1}$ be a smooth closed manifold, embedded into the round sphere $\mathbf{S}^n$ via a regular map $\Phi$. Using tools from geometric measure theory, one can prove the existence of a $n$-...
user avatar
  • 2,462
0 votes
0 answers
44 views

Are there references to functional variants of the Unbounded Knapsack Problem?

Looking for a version of the following problem, extended to solutions in $\ell^{\infty}(\mathbb{N})$ Unbounded Knapsack Problem $ \max_{x_1,...,x_n} \sum_{i=1}^n v_ix_i$ $\text{ subject to }$ $\sum_{...
user avatar
  • 175
1 vote
0 answers
87 views

Can min-max be set up around a minimal cone?

Let me state my question in very loose terms to start, then give some details and restate it in more precise terms at the bottom. Question. Given a regular minimal cone $\mathbf{C}$, can one set up a ...
user avatar
  • 2,462
1 vote
0 answers
67 views

Dirichlet-to-Neumann map for second order ODE

Problem statement In a problem of interacting particles, I encountered a type of geodesic equation in $\mathbb{R}^n$ with an additional rotation and dilation term $$ \ddot\gamma(t) + e^{t Q} \Lambda ...
user avatar
2 votes
0 answers
91 views

About Hilbert-Haar theory

The Hilbert-Haar theory says that functionals $\mathcal{F}(u,B)=\int_{B} F(\nabla u)\,dx$, where $F$ is a convex function and $B$ is a bounded domain in $\mathbb{R}^N$, take a minimum in the space of ...
user avatar
0 votes
0 answers
59 views

How can I show $ H^{1/2}(\partial\Omega;\mathbb{R}^m)\subset \operatorname{Trace}\left(H^{1}(\Omega;\mathbb{R}^m)\right) $ is true?

I am recently reading a book about elliptic PDEs. In the proof of a theorem, there is the following statement. Let $\Omega\subset\mathbb{R}^d $ be a bounded domain with Lipschitz boundary and let $ H^...
user avatar
3 votes
2 answers
157 views

Variational principle for relativistic gas dynamics

I know quite a lot of Variational principles (VP) yielding systems of classical mechanics. By a VP, I mean something like $$\delta{\cal L}[U]=0$$ where ${\cal L}$ is a functional and the field belongs ...
user avatar
  • 47.5k

1
2 3 4 5
10