# Questions tagged [variational-analysis]

Variational analysis is a branch of mathematics that extends the methods arising from the classic calculus of variations and convex analysis to more general problems of optimization theory, including topics in set-valued analysis, e.g. generalized derivatives.

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### Find Extremum of Functional with No Initial Conditions

I have the following functional $J(x)=\int\limits_{a}^{b}[{y(x)^2 + \dot{y}(x)^2 + 2 y(x) e^x}] dx$ and I want to find the extremum of it. So I compute the Euler-Lagrange Equation and I get the ...
0answers
28 views

### Given a smooth family of strictly convex norms in $\mathbb{R}^n$ ($n\geq 2$), what can we say about the family of dual norms?

Let $U\subset{\mathbb{R}^n}$ be an open set (may has compact closure) and $F:U\times\mathbb{R}^n\rightarrow\mathbb{R}$ such that $F(p,\cdot):\mathbb{R}^n\rightarrow\mathbb{R}$ is, for each $p\in{U}$, ...
0answers
44 views

### Minimizing elastic energy of shrinking balls

I am new to variational analysis and I am currently working in the following setup: We denote the $2$-sphere with radius $r>0$ by $S_r^2$ and $S^2:=S_1^2$. In coordinates $(\theta,\varphi)$, ...
1answer
33 views

### Calculus of Variations by Charles Fox: Question on Statement in Section 2.4

Fox states in Section 2.4, pg. 38, that "Anticipating this result, it follows that even if $u(x)$ vanishes at either or both of the values $x=a$ and $x=b$, both $t^2(a)/u(a)$ and $t^2(b)/u(b)$ still ...
1answer
84 views

### Difficulty Understanding Sufficient Conditions for Weak Extrema in Calculus of Variations

I am having a difficult time understanding Jacobi's necessary condition for weak extrema of functionals. Graphics and detailed explanations would be helpful. I am following the following two texts: ...
0answers
31 views

### Euler-Lagrange equations from a complex-valued Lagrangian

I've been looking without success for references describing a generalization of the Euler-Lagrange equations that would be derived from a complex-valued Lagrangian density. I realize that “minimum” ...
2answers
67 views

### Calculus of Variations (Gelfand & Fomin): Proof of Euler's Equation for Constrained Variation

I am in Section 2.12.1 of Calculus of Variations by Gelfand & Fomin. I am attempting to follow the proof of the Euler equation for Constrained Variation (Theorem 1, pg. 42). However, I'm confused ...
0answers
17 views

### Maximizing a nested sum of infinitely many variables (discrete variational calculus)

I am wondering if there is a way of finding a neat closed form solution for a series of numbers $r_t$ that satisfies the following. This is supposed to give the optimal strategy of an economic agent ...
1answer
61 views

0answers
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### Minimum expected value over all probability functions

Find the minimum value of $E[X]$ over all probability density functions f(x) satisfying the following three constraints: (I) $f(x) = 0$ for $x \leq 0$ (II) $\int_{-\infty}^{\infty} f(x) dx = 1$ ...
1answer
52 views

### Neumann problem with zero average sobolev space

Let us considère the following Laplace-Neumann problem $-\Delta u=0$ with homogenuous boundary condition of type neumann, i:e $\frac{{\partial u}}{{\partial n}} = 0$. The variational formulation is ...
1answer
48 views

### How to take derivative of integral of function?

I'm reading a textbook where it forms a Lagrangian function $$L = \int_0^1 f(x)^{1 - \frac{1}{\alpha}}dx - \lambda\int_0^1 g(x) f(x) dx$$ But how do you take the derivative of this thing? The ...
0answers
89 views

### Optimization in Banach space: Find functions that minimize the supremum of a convex operator.

Let $D \subset \mathbb{R}^n$ be compact. Denote by $C(D, \mathbb{R}^n)$ the space of continuous functions from $D$ to $\mathbb{R}^n$. Let $K$ be a real, symmetric, positive-definite $n \times n$ ...
0answers
37 views

### Intuition behind Pohozaev identity

I'm wondering if the Pohozaev identity: n\int_\Omega\int_0^{u(x)}f(t)\operatorname{d}t\operatorname{d}x-\frac{n-2}{2}\int_\Omega u(x)f(u(x))\operatorname{d}x=\frac{1}{2}\int_{\partial\Omega}\left|\...
1answer
328 views

### Confused by Kullback-Leibler on conditional probability distributions

I understand the Kullback-Leibler divergence well enough when it comes to a probability distribution over a single variable. However, I'm currently trying to teach myself variational methods and the ...
0answers
96 views

### Does this functional satisfies the Palais-Smale condition?

Let $\Omega$ be a non-empty bounded open subset of $\mathbb{R}^N$, $\lambda\in \mathbb{R}$ be an eigenvalue of $-\Delta$ on the Sobolev space $H^1_0(\Omega)$ and $f\in L^\infty(\Omega\times\mathbb{R})$...
1answer
82 views

### Characterisation of the rate distortion function: issue with functional derivative

In Elements of Information Theory, I can't figure out how the functional derivative $\frac{\delta J}{\delta q(\hat{x}|x)}$ for \$ J(q) = \sum_x \sum_{\hat{x}} p(x)q(\hat{x}|x)\log{\frac{q(\hat{x}|x)}{...