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Results tagged with calculus-of-variations
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user 119793
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
5
votes
3
answers
496
views
Uniqueness of minimizers in a problem in the Calculus of Variations - Part II
Take any convex set $A\subset\mathbb{R}^n$ which contains a neighborhood of the origin and let $f_A$ be the associated Minkowski functional
$$
f_A(x) = \inf\{\lambda>0\mid x\in\lambda A\},
$$
which tu …
- 391
3
votes
1
answer
165
views
Infimum of an integral functional involving a symmetric matrix
I have a symmetric $d \times d$ matrix $A$ and I have the following functional:
$$
\mathcal J(h) := \int_{B_1(0)} \vert \langle Au,u \rangle\vert \frac{\vert h'(\vert u \vert)\vert}{\vert u \vert} du, …
- 391
6
votes
1
answer
185
views
A one-dimensional integral minimization problem
Let $\mathscr F$ be the collection of smooth functions $f \colon
\mathbb R \to \mathbb R$ such that
$f \in C^\infty_c(\mathbb R)$, with $\text{supp } f \subset [-1,1]$;
$\int_0^1 x f(x) …
- 391
4
votes
1
answer
277
views
On convex functions which are non constant on every segment
I have been studying for the last few weeks the paper Dirichlet problem for demi-coercive functionals by Anzellotti, Buttazzo and Dal Maso. In this work the authors introduced and studied the variatio …
- 391
3
votes
1
answer
190
views
Uniqueness of minimizers in the Calculus of Variations
Let $f \colon \mathbb R^2 \to \mathbb R$ be the function defined by
$$
f(x,y):= (x^+)^2 + (y^+)^2
$$
where $a^+ = \max\{a,0\}$ for any real number $a$.
Given a Lipschitz regular domain $\Omega \sub …
- 391
3
votes
2
answers
188
views
Example of convex functions fulfilling a (strange) lower bound
I am reading a preliminary version of a paper which focuses on some minimization problems connected to a class of integral functionals. Reading the assumptions of one of the theorems I cannot convince …
- 391
4
votes
1
answer
155
views
Non-linear translation invariant functionals on $L^1$
I have recently come across a class of (possibly non-linear) operators $F$ defined on $L^1$ such that
$F \colon L^1(\mathbb R^d) \to \mathbb [0,+\infty]$;
$F(u(\cdot - z)) = F(u(\cdot))$ for every …
- 391