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-1
votes
0answers
40 views

Bounding sum of norms by the sum of sqaure of norms [closed]

How can you bound sum of norms (e.g. sum of norms of vectors) by sum of square of the same norms? Please advise. Thanks in advance.
2
votes
2answers
621 views

A difficult integral [closed]

Is there any analytical result on the following integral? $$\int_{-\infty}^{\infty} \frac{e^{-x^2}}{1+e^{-(x-\mu)}}dx$$ Thanks a lot!
0
votes
0answers
10 views

Is there any Monte carlo or statistical approach to variational integral problems?

I am just shooting in the dark: From brain data imaging we have integrals of the form $L(D):=\int_{\Omega}(\left \| A_{tensor}(D)-\widehat{A}\right \|+\sqrt{|\gamma(D)|})d\Omega$, where we minimize ...
0
votes
0answers
20 views

Measure of sub level of a torsion energy

Given a domain $\Omega$ (not necessarily open, but bounded. We can take quasi open domain). And let $u_{\Omega}$ be the minimizer of the torsion energy, $$ \int_{\Omega}|\nabla u |^2\, -\, \int_{\...
1
vote
1answer
143 views

Shannon's proof of the entropy power inequality

In Shannon's paper on information theory, found here, he asserts the entropy power inequality in appendix 6, found on page 52. I was reading his proof and it seems like there is a gap. Through his ...
1
vote
1answer
191 views

Question on Harmonic maps between Riemannian manifolds

In Theory of harmonic maps, main goal is to find minimum of Dirichlet energy function which is defined as follows: $$E(f):=\frac{1}{2}\int_M\|df\|^2dvol_g\qquad f:(M,g)\to(N,h).$$ In many Books such ...
4
votes
1answer
108 views

Calculus of variations when functional involves inverse of the function

Typically the Euler-Lagrange equations are defined for the functional $$ J[u] = \int_a^b L(x,u,u') dx. $$ However, I was wondering if anyone knows if they can be solved when the expression involves ...
2
votes
0answers
59 views

Currents with mean curvature

so let $T=\tau(M,\theta,\xi)$ a rectifiable current in $U\subset\mathbb{R}^{n+k}$, then $T$ has mean curvature vector $H$ if for every $X\in C^1_c(U\setminus\partial T,\mathbb{R}^{n+k})$ the variation ...
1
vote
0answers
50 views

First variation on double integral [closed]

Currently I am trying to fully understand the paper of munk1921. In the derivation of the minimum induced drag theorem it is at one point stated (p.378) that in order to minimize drag the following ...
0
votes
1answer
37 views

Existence of stationary tangent cones

My question refers to Leon Simon's Book: Geometric Measure Theory, Chapter 42 So let $V\in G_n(U)$ a general Varifold and $\|\delta V\|$ the Total Variation measure. If the density $\theta^n(\mu_V,x)&...
2
votes
0answers
44 views

decomposition of codim 1 currents

Let $T$ be an codimension one rectifiable mass-minimizing current. If $\partial T\llcorner B_R(a)=0$, then it is possible to write $T$ as an infinite sum of oriented boundaries: $T\llcorner B_R(a)=\...
2
votes
0answers
89 views

Willmore functional

Let $(M^2,g)$ and $(\bar{M},\bar{g})$ be two Riemannian manifolds. Suppose that $\mathcal{W}$ is the Willmore functional on the set of immersion functions from $M^2$ to $\bar{M}$. We know that $(\...
1
vote
1answer
54 views

Extremal of an L^1 continuous functional on a compact bounded set

Please, I need a small help with a reference. Lets say we do have a continuous functional $f$ on $L^1$ space and we want to prove the existence of extremals $f(\Omega)$, where $\Omega$ is compact and ...
5
votes
3answers
349 views

Euler-Lagrange equations and Bellman's principle of optimality

One method to optimize the integral $$\int_{\mathcal T} L(t,x,\dot{x}) \; dt $$ of a functional over a curve is the calculus of variations, which leads to ordinary differential equations: the Euler-...
3
votes
0answers
67 views

Invariant Lagrangians of a connection and its derivatives: how do they look like?

Let $$ L=L(\Gamma,\partial\Gamma,\ldots,\partial^n\Gamma) $$ be a Lagrangian depending on a linear symmetric connection $\Gamma$ on the tangent space of a manifold $M$ together with its derivatives up ...
4
votes
0answers
49 views

Why can every twist map be realized as the time-1 map of a time-dependent Hamiltonian?

if have problems getting my head around the following claim made by Moser in "Monotone twist mappings and the calculus of variations" and Gole in "Symplectic twist maps". Setting: Let $F : \mathbb{R}...
5
votes
1answer
142 views

An extension of Hadamard maximum determinant problem

Consider the Vandermonde product $\prod_{1\le j < k \le n} |z_j - z_k|$. It is well-known that under the constraint $|z_j| \le 1$ for all $j$, the product is maximized at a picket fence ...
0
votes
1answer
109 views

Generalized Lax-Milgram for Weak Formulation of 1D Linear Schrodinger

I am interested in the variational formulation of the 1D Schrodinger equation: $i u_t- \beta u_{xx} = 0 $ and $u(x,0)=u_0(x)$ which upon integration by parts yields: $i(u_t,v) + \beta (u_x,v_x) = 0$ ...
1
vote
1answer
97 views

The space of loops as a Banach space [closed]

Let $S$ be the space of all loops in $\mathbb{R}^2$, i.e. all continuous mappings $\gamma:[0,1]\to \mathbb{R}^2$ such that $\gamma(0)=\gamma(1)$. Is there any canonical interpretation of $S$ as a ...
2
votes
1answer
114 views

Almgren's mimeographed lectures notes on varifolds

I am trying to get some insights for the combinatorial argument of Pitts (in his PhD thesis 'Existence and regularity of minimal surfaces in Riemannian manifolds', Princeton University Press, 1981) to ...
2
votes
1answer
110 views

Reference Request: Variational Problem

I want to solve approximately the following variational problem: Given a function $c:[-1,1]^2\rightarrow [0,1]$, constants $p_1...p_n\in \mathbb{R}^+$, $\alpha_1...\alpha_n\in \mathbb{R}$, and $\...
2
votes
2answers
140 views

Total absolute variation of brownian motion, with different sampling rates

Let $(B_t)$ be a brownian motion on [0,1]. For the following, let $\omega$ be fixed. Let's compute the total absolute variation when sampling period = $\delta$ is fixed: $$V(\delta) = \sum_{i=0}^{N-...
1
vote
0answers
105 views

Interchange of integral and infimum

Can anyone please suggest how to justify widely used formula for interchange of integral and infimum: $ \inf_{u(t)\in U}\int_{t_0}^{t_1}g(t,u(t))dt=\int_{t_0}^{t_1}\inf_{u\in U}g(t,u)dt, $ where $ U\...
23
votes
2answers
925 views

How can you compute the maximum volume of an envelope(used to enclose a letter)?

It's obvious that the volume of a envelope is 0 when flat and non-0 when you open it up. However, if you were to fill it with liquid, there must be some shape where it has a maximum volume. Is there a ...
5
votes
0answers
141 views

a variational problem related to weighted logarithmic capacity

Consider the following multiple contour integral: $$ \Phi_\lambda := \oint \ldots \oint \prod_{1 \le j < k \le n} (z_j^{-1} - z_k^{-1}) \prod_{j=1}^n \prod_{k=1}^n (1 - z_j x_k)^{-1} \prod_{j=1}^n ...
1
vote
2answers
196 views

The term for problems “like” Brachistocrone?

Is there a commonly-accepted umbrella term for infinite-dimensional calculus problems where the goal is to compute an optimal geometric path between a pair of points? Three examples of this would be ...
1
vote
1answer
182 views

Is there a reason for different nomenclature on Calculus of Variations?

While sightseeing aspects of Calculus of Variations, the following fact elludes me: there is a plethora of new definitions which seem redundant to me. This phenomenom happens, of course, with other ...
1
vote
0answers
43 views

Numerical techniques for nonlinear, coupled integro-differential equations

The gist of the problem I have is I want to be able to find a numerical solution to these three coupled, rather unpleasant looking integro-differential equations (1): $$ \frac{d^2 x(t)}{dt^2} = \frac{...
0
votes
0answers
60 views

Well-posedness of gradient flows

For a convex lower-semicontinuous functional on a Hilbert space $I\colon H\rightarrow\mathbb{R}$, it is shown in Evans' PDE that the Hilbert-space-valued ODE $$\begin{cases}\mathbf{u}'(t)\in-\partial ...
0
votes
1answer
99 views

gradient descent in space of functions

Differential equations of the form $$\frac{d}{dt}\vec{x} = - \nabla E(\vec{x})$$ can be analyzed using phase portrait method. In particular, if the function $E$ (we call it energy) has local minimums, ...
4
votes
1answer
391 views

A continuous path between two Sobolev functions

Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $u_1$, $u_2\in H^{1}(\Omega)$ such that $T[u_1]=T[u_2]=T[\omega]$ where $T$ stands for the trace operator and $\omega\in H^1(\...
0
votes
1answer
118 views

Double Integral Equations

In my research I've come across a handful of double integral equations, and I'm nearly at a total loss for how to derive anything useful from such things. I've been lead to believe that even single ...
4
votes
1answer
202 views

Questions about the regularity of the “norm” associated to a convex set

Suppose $K\subset \mathbb{R} ^n$ is a closed convex set whose interior contains the origin. We can assign a gauge function to $K$ as $g_{K}(x):=\inf\{\lambda>0 \mid x\in\lambda K\}$. $g_K$ has all ...
5
votes
0answers
77 views

What is the maximal convex hull in $\mathbb R^3$ of a tree with fixed total length?

Denote by $\mathcal T_n$ the set of all trees on $n$ nodes. For a tree $T\in\mathcal T_n$, we assign to each edge a non-negative length such that the sum of all lengths is 1. Denote by $v(T)$ the ...
5
votes
0answers
104 views

Methods of variational calculus in analytic number theory

What methods of calculus of variations have been used in analytic number theory? I mean do Hamilton-Jacobi theory of PDE found usage in analytic number theory, which raises yet another question has ...
7
votes
2answers
311 views

Closed curve whose neighborhood is as large as possible

Let $C$ be a closed curve in the plane and let $N_\epsilon(C)$ be an $\epsilon$-neighborhood of $C$, like this: (ignore the fact that the "curve" is polygonal in this picture, I drew it in MATLAB) ...
4
votes
0answers
67 views

Minimisers and critical points of variational integrals

In the following we consider $\Omega\subset\mathbb{R}^n\ (n\geq2)$ to be open, bounded and with Lipschitz boundary. Consider the following regular variational integral: \begin{equation*} I[u]=\int_{\...
6
votes
1answer
309 views

electron configuration on manifolds

Let $M$ be a Riemannian manifold. For $k\geq 2$, suppose there are $k$ particles whose mass and volume can be regarded as zero and negatively charged with electricity equally. These $k$ particles move ...
1
vote
0answers
80 views

Existence of at least one positive solution for semilinear biharmonic equation with critical exponent

Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow \begin{cases} ‎\Delta^2u=‎‎\lambda ‎‎\dfrac{u}{|x|^4}‎‎+u^{‎p}‎ & \mathrm{in}‎\hspace{...
4
votes
1answer
186 views

Fair surfaces - general mathematical theory

Fairness measures for surfaces are, in general, functionals containing more complicated terms thatn the usual bending energy, and may depend not only on the mean curvature but also on principal ...
3
votes
0answers
68 views

Can Mumford-Shah functional be adapted to lower $L^1$ space?

The well know Mumford-Shah functional functional $$ F(u)=\int_\Omega|\nabla u|^2+\mathcal H^{N-1}(S_u) \tag 1 $$ where $u\in SBV(\Omega)$ and $\nabla u$ is the absolutely continuous part of ...
2
votes
0answers
22 views

Is this function $u\in SBV(\Omega)$ also belongs to $L^\infty(\Omega)$?

Some early discussion can be found here. My question: Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Also $\mathcal H^{N-1}(\partial\Omega))<\infty$. Let $u\in SBV(\...
0
votes
0answers
49 views

Nonlinear optimization problem with inequality constraints

Consider a real valued function $g(x_i)=\frac{1}{a_1+ \frac{a_2}{x_i}}, \forall i=\{1,2,3,...,n\}$. The objective function $H$ is $H=\sum_{i=1}^{n}\frac {1}{g(x_i)-a_3x_i}$ The optimization ...
1
vote
1answer
81 views

question about $TGV^2$ space

Let us just stay in $\mathbb R^1$. The space $TGV^k$ is defined as the function $u\in L^1(I)$ and $$ TGV^k(u,I):=\sup\left\{\int_I u\,\phi^{(k)}\,d\mu, \,\phi\in C_c^\infty(I),\,\|\phi\|_{L^{\infty}(...
0
votes
0answers
120 views

Is this has anything to do with Riesz representation?

The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49. Here I come up with a question which has similar ...
0
votes
1answer
98 views

Constrained optimal control problem

I am trying to solve an optimization problem which is probably reminiscent of optimal control theory but all of this is not exactly my field of expertize and I am a little bit lost in translation. If ...
1
vote
1answer
147 views

Is it true that for dimension d=3, if $v\in H_0^1(\Omega)$ but $v\notin L_\infty(\Omega)$ then exponent of v or -v is not summable?

I have the following Question: 1) Is it true that if $\Omega\subset\mathbb R^3$, $\Omega$ - bounded, $v\in H_0^1(\Omega)$ but $v\notin L_\infty(\Omega)$ implies $\int\limits_{\Omega}{e^vdx}=+\infty$ ...
0
votes
1answer
134 views

How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?

Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary. My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that $...
8
votes
1answer
202 views

Which domain maximizes the energy of the Lebesgue measure?

This could be asked in more generality, but let me stick to a concrete case. Usually one considers a fixed domain $E \subset \mathbb{C}$ and attaches to it the equilibrium probability measure $\nu_E$...
3
votes
0answers
60 views

Morse index in variational calculus

Let $f(t,x,v)$ be a smooth function on $\mathbb{R}\times \mathbb{R}^n\times \mathbb{R}^n$, $\mathcal{C}_{x,y}$ be the set of smooth paths $c:[0,1]\to \mathbb{R}^n$ with $c(0)=x,c(1)=y$, $E(c)$ be a ...