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0
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1answer
81 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 ...
0
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0answers
20 views

Euler-Lagrange equation with constraints outside the integral [migrated]

So i've been studying Euler-Lagrange equations, and on an assignment i have the problem to find them for $J(y)=\int_a^bF(x,y,y')dx-By(b)+Ay(a)$ Where y(a) and y(b) are free, A and B are constant. ...
0
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0answers
37 views

Is the following definition of a functional derivative natural?

if $\delta S = \int \sqrt g F[\phi] \delta \phi$ Then is it natural to define the functional derivative as follows, $\frac{\delta S}{\delta \phi} = F[\phi]$. In particular does this definition ...
8
votes
1answer
165 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 ...
3
votes
0answers
41 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 ...
0
votes
1answer
97 views

Lagrangian flow preserves symplectic form

Let $X$ be a configuration space and $L: TX \rightarrow \mathbb{R}$ a Lagrangian. Then I want to show that the Lagrangian flow $F^t(x(0),x'(0)) = (x(t),x'(t))$ preserves the symplectic form just like ...
6
votes
1answer
221 views

Functional minimization problem

Is there a smooth solution to minimize this: $$ \int_0^1{x \over {1+k^2f'(x)^2}}dx, f(0)=1, f(1)=0, f'(x)\leq 0, k^2>0. $$ I could "solve" it using a numeric approximation (my algorithm converged ...
8
votes
3answers
514 views

Rigorous justification that overdetermined systems do not have a solution

There is the following well known and very useful heuristic principle: Assume one has a natural map from the space of $k$-tuples of functions in $n$ variables into the space of $K$-tuples of functions ...
3
votes
1answer
362 views

A priori estimates for a nonlinear elliptic problem singular on the boundary

Let us consider the following elliptic problem $$ \begin{cases} -\Delta u = \frac{u^p}{|x|^2} \mbox{ in } \Omega \\ u >0 \mbox{ in } \Omega \\ u = 0 \mbox{ on } \partial \Omega. \end{cases} $$ with ...
0
votes
0answers
79 views

Functional involving Ricci curvature: convex and coercive?

Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$, volume form $\mu_g$, and Ricci curvature $\text{Rc}_g$. Question: Given a fixed vector field $V\in\Gamma(TM)$, under what ...
1
vote
1answer
270 views

Euler-Lagrange Equation and “Eigen Value ”

I posted this question on Math.SE, but I could not get any help. The eigenvalue $\lambda(t)$ is characterised as the minimum of the Rayleigh quotient (where $t$ is a scalar variable) ...
0
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0answers
62 views

About bounding values of quadratic forms

It would be helpful if someone can share (either as references) examples of calculations/analysis which achieves bounding of values of quadratic forms in say either of the following situations, ...
3
votes
1answer
162 views

Gross's log Sobolev inequality proof with variational calculus?

For $f\in C^{1}(\mathbb{R}^{n})$, Gross's logarithmic Sobolev inequality says that $$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\int f^{2}\,d\mu\right)\leq \frac{2}{c}\int |\nabla ...
0
votes
0answers
36 views

A special approximation of BV functions

Suppose $\Omega:=B(0,1)\subset \mathbb R^2$. Let $u\in BV(\Omega)$ be a radially symmetric, i.e., $u(x)=u(Rx)$ for all $R\in SO(2)$. In addition, suppose $w$ to be an affine function,. i.e., ...
6
votes
1answer
108 views

Inequality of the norm of the convolution in $L^p(\mathbb{R}^n)$ with symmetric decreasing rearrangement?

Is it true that $$ ||f*g||_p \le ||\,|f|^* * |g|^*||_p\quad ? $$ where $|f|^*$ and $|g|^*$ are the symmetric decreasing rearrangements of the functions $|f|$ and $|g|$. Under what conditions on $f$ ...
1
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0answers
64 views

Method of proving the regularity of the minimizer of geometric variational problems

Recall the proof of the $(\Lambda,r_0)$-perimeter minimizer. We say that $E$ is a $(\Lambda,r_0)$-perimeter minimizer in an open set A provided $sptD\chi_E=\partial E$ and there exist two constants ...
7
votes
1answer
149 views

Mountain Pass theorem for minimization problems with constraints

Let $I[u]$ be a functional on a (possibly infinite dimensional) Hilbert space. Then, under some conditions, the Mountain Pass theorem guarantees the existence of a saddle point (see ...
0
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0answers
34 views

Interesting properties of complex Gateaux derivatives

The complex Gateaux derivatives are for $\psi=\psi_{1}+i\psi_{2}$ and functional F $dF(\psi,\xi)=d_{\psi_{1}}F(\psi_{1},\xi_{1}-id_{\psi_{2}}F(\psi_{2},\xi_{2}$ and ...
0
votes
0answers
78 views

Showing existence of positive weak solution of a PDE by CoV

Given the following PDE $$ \begin{cases} -\Delta u+\alpha=u^q &x\in\Omega\\ u=0 &x\in\partial\Omega \end{cases} $$ where $\Omega\subset\mathbb R^3$ is open bounded with smooth boundary, ...
4
votes
1answer
231 views

Calculus of variation

This is probably simple but I'm stuck somewhere. I am trying to solve the calculus of variation problem that arise in an applied field: $$\min_{f \in C^1} \int^1_0 \int^1_0 (x-y)^2f(x,y)dxdy$$ ...
6
votes
1answer
187 views

Survey paper on isoperimetry

I am searching for a comprehensive survey article (or more different articles) on the subject of isoperimetric problems from ancient Greece to modern mathematical physics. Could you point out some ...
1
vote
0answers
85 views

Free Endpoint of Minimization Problem

Consider the following minimization problem $$\inf \left\{ \int\limits_{-\infty}^0 \left[ (\psi')^2 + m(y)(\psi - F)^2 \right]\; : \; \psi \in H^1(\left(-\infty,0\right]) \right\}$$ where $m(y) > ...
1
vote
1answer
53 views

Convergent algorithm for dividing a body into two regions of equal volume

Let $\Omega \subset R^3$ be a bounded open region. It is well known that there exists a smooth surface $\Gamma$ with minimum area and constant mean curvature which is orthogonal to $\partial \Omega$ ...
5
votes
2answers
214 views

Stability of minimal surfaces

Let $\Gamma$ be a prescribed $n-2$ dimensional set and assume $S \subset R^n$ is a minimal hyper-surface with respect to some smooth metric $g$ on $R^n$, and $\partial S= \Gamma$. Is $S$ is stable ...
9
votes
0answers
108 views

Area of the minimal surface of a non-planar quadrilateral in 3d

Consider a non-planar quadrilateral in three dimensions, i.e. four points $x_1,\dots,x_4$ in $\mathbb{R}^3$ that do not lie on a plane and connected by straight lines. Then, by general theory of ...
6
votes
0answers
173 views

What is the first eigenvalue of $p$-Laplacian on unit sphere $S^n$?

We know that the first eigenvalue of Laplacian on the Riemannian unit sphere $S^n$ is $n$, then what is the explicit expression for the first eigenvalue of $p$-Laplacian on $S^n$? The $p$-Laplacian ...
4
votes
0answers
95 views

Sobolev spaces of maps between manifolds and the Palais-Smale Condition

I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts: Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean ...
0
votes
0answers
72 views

Integral representation of the function

I try find integral representation to following function ($d_1<d_2$ and $d_2-d_1=1(\mod 2)$ $f(d_1,d_2)=\frac{(-1)^{\frac{d_2-d_1-1}{2}}}{2^{d_1-1} d_1!}\sum_{k=[\frac{d_1}{2}]+1}^{ d_1} { d_1 ...
3
votes
5answers
480 views

Variation of curvature with respect to immersion?

Let $M$ be a smooth surface and let $f: M \to \mathbb{R}^3$ be a family of immersions given by $$ f(t) = f_0 + tuN_0, $$ where $f_0$ is some initial immersion, $N_0$ is the associated Gauss map, and ...
14
votes
3answers
559 views

Central limit theorem via maximal entropy

Let $\rho(x)$ be a probability density function on $\mathbb{R}$ with prescribed variance $\sigma^2$, so that: $$\int_\mathbb{R} \rho(x)\, dx = 1$$ and $$\int_\mathbb{R} x^2 \rho(x), dx = \sigma^2$$ ...
0
votes
0answers
31 views

Standard Arguments of Calculus of Variations [duplicate]

I am working on calculus of variations in solid mechanics. I did my studies in Civil Eng., so I haven't passed any courses on Math Analysis. I do have problems with main properties of Hilbert and ...
6
votes
1answer
198 views

Generalizing “variation of parameters”

I'm stuck on generalizing an ODE formula and could use your help! One way to think about "variation of parameters" is that it bakes the solution $z(t)=e^{At}z_0$ of $z'=Az$ (here ...
0
votes
0answers
64 views

numerical and functional mixed optimization problem $\max f - \min f +\int_{-1}^1 (f'(x)-x)^2dx$

Given a function $g(x)$ and its domain, we want to get another function $f(x)$ whose derivative is approximately $g(x)$, but so that $f(x)$ itself has small variation. For example, for ...
1
vote
0answers
73 views

Maximizing the “uniformity” of a probability measure, with constraints, via path length minimization

Background I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform ...
0
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0answers
11 views

Complex parameters in the Ritz procedure

I am using the Ritz procedure to write a trial function as the superposition of other admissible functions, with the coefficients being unknown variational parameters to be determined. The variational ...
3
votes
0answers
59 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
134 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
136 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
90 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
192 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, ...
23
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2answers
4k views

The error in Petrovski and Landis' proof of the 16th Hilbert problem

Edit:For a recent progress on the Hilbert 16th problem see the following note which consider an infinite dimensional nature for this (apparently 2 dimensional) amazing problem . Best wishes for ...
2
votes
0answers
100 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\} ...
3
votes
1answer
77 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
271 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
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0answers
146 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
402 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
105 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
203 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
191 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
105 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 $y_\Omega(0) = a_\Omega, y_{\Omega}(1) = b_\Omega$. ...