2
votes
0answers
48 views

Prove that the maximizing point configuration on the unit circle for a Vandermonde like functional is a picket fence

This is probably too hard for math.stackexchange, so I migrated it here. For $\lambda_i \in S^1 \subset \mathbb{C}$, consider the functional $H(\{\lambda_1, \ldots, \lambda_n\}):= \sum_{j < k} | ...
2
votes
1answer
99 views

Projection onto $\ell^{2,1}$ ball

Does anyone have an idea how to project onto the $\ell^{2,1}$ ball efficiently, i.e. how to solve $$ u = \arg \min_u \|u-f \|^2 \text{such that } \left(\sum_i \big(\sum_j |u_{i,j}|\big)^2 ...
0
votes
0answers
116 views

How to prove this inequality of heat flow from Weitzenbock formula?

Let $(M,g), (N,h)$be a compact Riemannian manifolds, $m:=\dim M, n:=\dim N\geq 2$, and $N$ is a non-positive curvature $K_N\leq 0$. All connections which appear below are the Levi-Civita connections. ...
3
votes
0answers
74 views

Variational Principle for a System of Differential Equations

I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v ...
3
votes
1answer
103 views

Find a maximizing solution to an ODE which depends on a paramater function

(For the physical meaning of this problem see http://physics.stackexchange.com/questions/122818/how-should-i-throttle-my-rocket-to-reach-highest-altitude). Given $g \in (0,\infty), k \in C^1( [0, ...
2
votes
0answers
82 views

Minimizing some $H^{-1}$ functional over (a subset of) probability densities in $R^d$

Let me consider the following subset of probability measures in $R^d$ $$ \mathcal{K}_M=\left\{0\leq u(x)\in L^1(R^d):\quad \int u(x)dx =1,\,\int|x|^2u(x)dx\leq M,\,\int u(x)|\log u(x)|dx\leq M\right\} ...
2
votes
1answer
72 views

Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
3
votes
2answers
233 views

Existence of Minimizer of $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $

given the following functional $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $ with $\rho>0$ , $\|\rho\|_1 = 1$ and obviously $\rho\in L^1(\mathbb R^3)$. Can I see ...
0
votes
0answers
114 views

Third variation of area of a minimal surface

There is a formula for the third variation of area on page 96 of Nitsche's book, Lectures on Minimal Surfaces, vol. 1 (English version). He says at the bottom of the page it is good for normal ...
10
votes
1answer
279 views

Who first resolved Hilbert's 20th problem?

Hilbert's 20th problem concerns the existence of solutions to the fundamental problem in the calculus of variations. I understand that Hilbert, Lesbesgue and Tonelli were pioneers in this area. In ...
2
votes
0answers
79 views

How can I find the spectrum of this operator?

I've posted this now in on /r/math on math.se to no avail. Maybe this problem is harder than I thought and you folks can help me out. I'm working on a variational problem in elasticity which ...
2
votes
4answers
153 views

Showing coercivity condition for an energy functional

Consider the energy functional $e(\cdot)$ \begin{align*} e(f,Q)&=\int_a^b \bigg\{f^4\bigg[1+\|\frac{d}{dr}Q\|^2+f^2\dot f^2\bigg]\bigg\} \,dr, \end{align*} over the space of \begin{equation*} ...
4
votes
1answer
155 views

Lieb: Stability of matter, problem with variational method

I am trying to recalculate 'Stability of Matter' Paper from Lieb. I have a problem at the last step of the proof at page 3, where Lieb calculates the minimizer. I will explain my ideas and my problem ...
1
vote
1answer
67 views

Implicit function theorem for boundary value problems

I have a nonlinear, two point boundary value problem of the form $F(x, y(x), y'(x); \Omega ) = y''$ along with some boundary conditions of the form $y_\Omega(0) = a_\Omega, y_{\Omega}(1) = b_\Omega$. ...
1
vote
0answers
149 views

Reverse Holder Inequality and the higher integrability of the gradient of a solution to Euler's equation for a certain functional

In Giaquinta-Giusti's (1978) paper "Nonlinear Elliptic Systems with Elliptic Growth" (thm 1.1) they consider the following system: \begin{equation} \sum_{i, j=1}^{n}\sum_{\alpha, ...
2
votes
1answer
104 views

Fréchet derivative of the Total Variation norm

Good day. The question is on proving the following relation ($\|\cdot\|$ here and on denotes $\ell_2$ norm): $$ \frac{dJ(f)}{df} = -\mathrm{div}\left(\frac{\nabla f}{\|\nabla f\|}\right) \qquad =: DJ, ...
6
votes
0answers
149 views

Choquet theory and Hilbert's fourth problem

The following text is an attempt to see Hilbert's fourth problem in a new light. Definition. A pseudometric $d$ on $\mathbb{R}^n$ is called projective if whenever a point $z$ belongs to a line ...
1
vote
0answers
66 views

monotone parabolic systems, convex variational structure and Legendre transform

The context: for my research I am currently looking at parabolic systems of the type $$ \left\{ \begin{array}{ll} \partial_t b(u)-\Delta u=0 \qquad & (t,x)\in \mathbb{R}^+\times\Omega\\ u=0 & ...
1
vote
1answer
69 views

variational problem related to an integral

Recently I came up with a type of variational problem in stochastic process. It can be stated in the following way: Given $a$ and $b$ positive, and an increasing function $f$ on $(0,1)$ (may be not ...
2
votes
0answers
63 views

Variational problem for optimal weight function leading to shorter intervals with many primes

The motivation for the following problem stems from the recent preprint by James Maynard, see also Proposition 5 of the recent blogpost by Terrence Tao. The solution of this problem could give better ...
4
votes
1answer
357 views

Failure of Palais-Smale Condition C and the Mini-Max Principle

To get a thorough analysis of the critical point structure of a smooth function $f:M\to\mathbb{R}$ on a smooth Hilbert manifold $M$, a compactness assumption gets us far. That assumption is Condition ...
0
votes
0answers
150 views

Can a function be constructed from the direction of its gradient?

Let $\Omega$ be a bounded region in $R^n$ and $J\in (L^2(\Omega))^n$ with $|J| \leq 1$ a.e. in $\Omega$. Under what conditions the equation $Du=J|Du|$, $u|_{\partial \Omega}=f$ has a solution in a ...
1
vote
1answer
226 views

Sobolev Inequality

Let $\Omega$ be a bounded region in $R^n$ and define $W:=\{ u \in H^{1}(\Omega): u(x_0)=0 \},$ where $x_0 \in \partial \Omega$ is a fixed point. Is there a constant $C$ such that ...
2
votes
3answers
271 views

Symmetry Properties of Minimizers - Calculus of Variations

What methods are there to show symmetry properties of the minimizer of a problem $\inf_{u\in X}\mathcal{F}(u)$ in the calculus of variations? In general, the symmetry properties of $\mathcal{F}$ do ...
0
votes
0answers
42 views

fundamental optimal-trajectory result known?

It's well-known and obvious that if you have a spaceship and your sole constraint is an upper bound on magnitude of acceleration/deceleration, the fastest way to get to a distant star (a fixed ...
2
votes
1answer
123 views

Is vesica piscis a maximal length curve constrained to two points?

Let $r(t), t\in [0,1]$ be a continuous piecewise $C^1$ curve on the plane where $r(0)=(0,0)$ and $r(1)=(1,0)$. The distance $|r(t)|$ is a non-decreasing function and the distance $|r(t)-(1,0)|$ is a ...
2
votes
1answer
247 views

Upper bound on the maxima of ratio of expectation of quantities under Gaussian measure

Let $\lambda,\eta >0$ be given, and $u:\mathbb{R}\rightarrow \mathbb{R}$ be a real valued function. Define $$\Delta(u)= \frac{\int u(h) \exp(-\eta ...
1
vote
0answers
127 views

Equal maximum and minimum in a large-scale linear programming

For a linear optimization of an integral (with integral constraints), I perform a linear programming for the equivalent series. Maximum and minimum of the LP problem tend to be equal as I increase the ...
0
votes
0answers
88 views

Reference Search for a Functional Minimization Problem

Let $u(x) \ge 0$ be a non-negative, piecewise-differentiable function on the real line. Moreover, let $u(x)$ be integrable with fixed positive mass, that is $$M \equiv\int_{x=-\infty}^\infty u(x) ~ ...
4
votes
1answer
120 views

Deriving Helfrich's shape equation for closed membranes

I have a bunch of papers that claim that, from the equation for shape energy: $$ F = \frac{1}{2}k_c \int (c_1+c_2-c_0)^2 dA + \Delta p \int dV + \lambda \int dA$$ one can use "methods of variational ...
4
votes
2answers
387 views

The Isoperimetric problem for domains constrained to lie between two parallel planes

It is well known that for a given volume $V$, a sphere is the shape that minimizes the surface area. I am interested in the same problem under the constraint that the shape must lie between the planes ...
3
votes
3answers
903 views

Good book on Calculus of Variations

What is a good book on the Calculus of Variations, for a second year PhD student?
2
votes
1answer
153 views

Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism

The variational bicomplex seams to provide a modern formulation of the variational problem in terms of modern differential geometry. In particular the bigraded complex of differential forms ...
2
votes
0answers
102 views

coordinate free Euler-Lagrange

The variational approach is to seek critical points in terms the Euler-Lagrange variational derivatives $E_a(S_0)$ of a local function $S_0.$ The zero locus does not depend on coordinates. Where is ...
1
vote
1answer
122 views

variational characterization of the average of an $L^p$ function

Let $\Omega$ be a measurable set having finite Lebesgue measure. Let $p\geq 1$ and $u\in L^p(\Omega)$. Is it true that the minimum value of the real function $$ c\in \mathbb{R}^n\mapsto\int_\Omega ...
7
votes
1answer
307 views

Invariance of the l.h.s. of Euler-Lagrange equation

Let $M^n$ be a smooth manifold equipped with a nondegenerate Lagrangian $L:TM\to\mathbb R$, $L=L(x,y)$, $x\in M$, $y\in T_xM$. The stationary points of the corresponding integral functional on curves ...
0
votes
1answer
156 views

Is still it weakly continuous ?

If $\{u_n\}$ is bounded in $H$(real Hilbert space)with inner product such that $(\cdot,\cdot)$, then ${\|u_n\|^2u_n}$ is bounded also. Passing a subsequence, one has that $\{\|u_n\|^2u_n\}$ converges ...
2
votes
0answers
62 views

Approximating solutions to minima of the discrete Lagrangian

I have been stuck on this problem for a week and I'm not sure whether or not it is hard or I'm just missing something obvious. General gist of the problem I have a variational problem on a ...
3
votes
1answer
216 views

Are all null curves of a Lorentzian metric extrema?

Suppose $g$ is a Lorentzian metric on $\mathbb{R}^4$, then consider the variational problem of finding extrema of $$F(\gamma) = \int_a^b \| \dot{\gamma}(s) \|_g ds$$ The so-called "null curves" are ...
0
votes
0answers
141 views

Extremals of functionals

Find the extremals of the functional $$J(x(t)) = \int\limits_{0}^{2}\frac{[x'(t)]^2}{x^3(t)}dt$$ subject to $x(0) =1$ and $x(2) = 4$. Does the two point boundary problem has a unique solution? Find ...
0
votes
0answers
72 views

Class of integrable 0/1-functions “with no null sets.”

I am looking for (the name of) a class of functions from $\mathbf{R}^2$ $(\mathbf{R}^n)$ to {0, 1} that are integrable. Let $f$ be in this class and $E$ be the set of all points where $f$ is equal to ...
4
votes
0answers
222 views

Symmetric matrices and Hilbert's fourth problem

From the analytic viewpoint, the Busemann-Pogorelov solution of Hilbert's fourth problem is summarized in the following result: Theorem. All straight lines are extremals of the variational problem $$ ...
7
votes
1answer
265 views

Formulating the calculus of varations with exterior calculus

I noticed that a calculus of variations problem is just an integral over a differential form. Therefore, I would think it would be possible to formulate the Euler-Lagrange equations using exterior ...
1
vote
1answer
258 views

Reversibility vs geodesic reversibility for Finsler metrics on the two-sphere

Problem. To give a concrete example of a geodesically reversible Finsler metric on the two-sphere that is not just the sum of a reversible Finsler metric and an exact $1$-form. Some background may be ...
1
vote
1answer
244 views

A 'conjecture' on critical elliptic pde

I conjecture the following. Let $\Omega=\mathbb{R}^3-\overline{B_1(0)}$. Define $$E_{\Omega}(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^2dx-\frac{1}{6}\int_{\Omega}|u|^6dx.$$ $E_{\mathbb{R}^3}$ is defined ...
0
votes
1answer
150 views

Boundary Problem with an Area Constraint

Consider a boundary given by vertices (0,a), (0,0) and (1,0) (an 'L' shaped boundary). The problem is to find the equation that passes between the endpoints (0,a) (1,0) of minimum length that ...
7
votes
1answer
244 views

Characterizing maximal powers of closed 2-forms in odd-dimensional manifolds

Given a nowhere-zero, closed $2n$-form $\Omega$ in a manifold of dimension $2n +1$, how do we know if there exists a closed $2$-form $\omega$ such that $\Omega = \omega^n$? Remark. This question ...
11
votes
2answers
524 views

A riemannian manifold with finitely many closed contractible geodesics

By a closed geodesic, I mean a smooth periodic geodesic $\mathbb{R} \rightarrow (M,g)$. I will consider them up to geometric distinction. This means that any two closed geodesics are equivalent if ...
0
votes
1answer
346 views

proving the existence of a minimizer of this functional

I seek the minimum of a certain functional which is always strictly greater than zero. The Euler-Lagrange equation is a "zeroth-order differential equation", that is, an implicit equation that a ...
5
votes
1answer
151 views

Minimizing the perimeter around an obstacle

Let $A\subset \mathbb{R}^n$ be a (measurable) bounded set, and consider the following optimization problem: minimize $P(X)$, the perimeter of a set $X$, where $X$ ranges over all Caccioppoli subsets ...