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 ...
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\} ...
3
votes
1answer
74 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
236 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 ...
4
votes
1answer
156 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 ...
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, ...
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 ...
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) ~ ...
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 ...
1
vote
1answer
266 views

How to explain the condition (C) in critical point theory?

Condition (C). The closure of any nonempty subset S of H on which f is bounded but on which $\|\nabla f\|$is not bounded away from zero, contains a critical point of f. How to see the meaning of " ...
0
votes
0answers
123 views

variational problem under convexity constraints

I wonder if there is any method to compute variational problems subject to certain shape constraints (e.g., convexity, monotonicity, etc.). The literature I found on this topic (which I am no expert ...
2
votes
2answers
336 views

Functional Minimization: When is this heuristic rigorous?

I'm trying to solve a functional minimization problem of the following form: $$\arg\min_{f:\mathbb{R}\rightarrow [0,1]} h(f)$$ where $h$ is some expression in terms of several integrals over $f$. I ...
2
votes
1answer
435 views

Convergence of mountain pass solutions of $-\Delta u+u=u|u|^{p-2}$

Consider the following equation in $\mathbb{R}^N, N \ge 3$: $$ (E) \quad -\Delta u +u=|u|^{p-2}u, $$ where $2 < p < 2^{*} =2N/(N-2)$. Denote by $J: H^1(\mathbb{R}^N) \to \mathbb{R}$ the ...
3
votes
1answer
1k views

Proof of a concentration compactness lemma

Hi I'm stuck with the proof of a concentration-compactness lemma. We have the following equation in $\mathbb{R}^N, N \ge 3$: $$ -\Delta u +u=|u|^{p-2}u, $$ where $2 < p < 2^{*}$. The functional ...
1
vote
1answer
546 views

Stuck on a convergence argument in $H_0^1(\Omega)$.

I'm trying to verify that a functional I have satisfies the Palais Smale condition for appliction of the Mountain Pass lemma. However I've encountered this step along the way which seems clear to me ...
8
votes
4answers
901 views

Boundedness of nonlinear continuous functionals

Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$. Is it true that there exists an infinite dimensional reflexive subspace $E$ of $C[0,1]$ s.t. $f(K\cap E)$ is bounded ? ...
3
votes
3answers
871 views

Minimizing a functional

I have wondered the problem in http://www.helsinki.fi/~hmkokko/Stuff/Esdale/index.html for over year without success. If we try to minimize the functional equation T(\theta ) = \int_0^L\frac ...