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1
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31 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}‎ & ...
4
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1answer
172 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 ...
2
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0answers
42 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
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0answers
12 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 ...
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0answers
40 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
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1answer
70 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 ...
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0answers
111 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 ...
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0answers
57 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 ...
0
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0answers
45 views

How to modify a SBV convergence sequence to obtain uniform integrability?

Given $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. Assume $(u_n)\subset SBV(\Omega)$ a sequence of functions such that $u_n\to u_0$ weakly in $SBV$ for some function $u_0\in ...
1
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1answer
123 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
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1answer
103 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
42 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
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1answer
179 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
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0answers
45 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
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1answer
99 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
241 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 ...
9
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3answers
531 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 ...
4
votes
1answer
392 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
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0answers
80 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
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1answer
271 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) ...
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0answers
64 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
185 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 ...
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0answers
42 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
113 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
75 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
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1answer
162 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
36 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 ...
1
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0answers
90 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
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1answer
234 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
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1answer
190 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
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0answers
86 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
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1answer
54 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
217 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 ...
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0answers
116 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 ...
9
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1answer
226 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
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0answers
99 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 ...
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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
489 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
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3answers
609 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 $$\int_\mathbb{R} x^2 \rho(x), dx = \sigma^2$$ ...
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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
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1answer
213 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 ...
3
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0answers
65 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
141 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 ...
3
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0answers
92 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 ...
4
votes
1answer
209 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, ...
25
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2answers
4k views

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

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis? Please see this related post and also the following post. Added : According to their ...
2
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0answers
101 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
275 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
151 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 ...