The calculus-of-variations tag has no wiki summary.

**2**

votes

**1**answer

155 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

**0**answers

104 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 ...

**9**

votes

**1**answer

245 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

**0**answers

67 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

**3**answers

117 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

**1**answer

142 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

**1**answer

51 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

**0**answers

82 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

**1**answer

96 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

**0**answers

131 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

**0**answers

57 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

**1**answer

63 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

**0**answers

59 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 ...

**3**

votes

**1**answer

338 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

**0**answers

148 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

**1**answer

196 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

**3**answers

252 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

**0**answers

41 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

**1**answer

119 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

**1**answer

227 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

**0**answers

118 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

**0**answers

84 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

**1**answer

108 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 ...

**0**

votes

**0**answers

155 views

### Variational Problem v.s. Initial Value Problem

Is there a way to relate the variational problem where one specifies $x$ initially and finally to the initial value (Cauchy) problem where one specifies both $x$ and $p$ initially? As an example, ...

**4**

votes

**2**answers

369 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

**3**answers

634 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

**1**answer

137 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

**0**answers

97 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

**1**answer

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

**1**answer

298 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

**1**answer

155 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

**0**answers

60 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

**1**answer

193 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

**0**answers

119 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

**0**answers

71 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

**0**answers

217 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

**1**answer

246 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

**1**answer

239 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

**1**answer

233 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

**1**answer

149 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

**1**answer

243 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 ...

**10**

votes

**2**answers

507 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

**1**answer

311 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

**1**answer

149 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 ...

**4**

votes

**1**answer

555 views

### Geometric derivation of the Einstein’s field equation from the Hilbert action.

It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional is given by the Einstein field equation (for a statement, see previous question). The standard derivation ...

**7**

votes

**1**answer

457 views

### Coordinate-free derivation of the Einstein's field equation from the Hilbert action.

It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional (without matter and cosmological constant, which is irrelevant here):
$$S = \int_M R \mu_g,$$
is given by ...

**8**

votes

**1**answer

408 views

### Vector fields on path spaces

I've been reading Chen's original works on iterated integrals and in order to consider differential forms on the path space $PM$ of a smooth manifold $M$ he gives $PM$ the following "differentiable ...

**10**

votes

**0**answers

188 views

### “Small” maps from sphere to sphere

Start with a continuous map $f:S^{n+k} \rightarrow S^n$ (round unit spheres). The graph of $f$ lives in $S^{n+k}\times S^n$ and suppose it has a surface area (as a subspace of co-dimension $n$). Now ...

**6**

votes

**3**answers

577 views

### Who came up with the Euler-Lagrange equation?

Which man came up with the solution to the basic Calculus of Variations problem first?
http://en.wikipedia.org/wiki/Euler-Lagrange
Makes it sound like Lagrange got it first in 1755, then sent it to ...

**4**

votes

**1**answer

229 views

### Work on an Einstein-Hilbert type action but with the *absolute value* of scalar curvature?

This is only my second question on mathoverflow, so my apologies if this would be more appropriate at a physics site. My question concerns a modification to the Einstein-Hilbert action. The standard ...