# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 " ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ? ...