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Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
1
vote
Lagrangian independent of derivative with a salvage value
Without controlling the derivative, the problem is a bit ill posed, as $x(T)$ can take any value independent of what $x$ does in the rest of the integral. So your solution will be a minimizer of $\int …
1
vote
0
answers
12
views
On the relation between quasiconvex functionals and quasimonotone operators
The following is a classical definition due to Morrey: Let $\Omega \subset \mathbb{R}^n$ be a nice enough, bounded domain and $f: \mathbb{R}^{m \times n} \to \mathbb{R}$ with some reasonable growth co …
1
vote
Accepted
Non convex optimization problem in $W_0^{1,2}$
You can treat this as a problem with two Lagrange multipliers. Then by standard methods, a minimizer $f$ has to exist (by convexity in $f'$) and has to be a weak solution to
$$-f'' + \lambda f + \mu f …
2
votes
Existence of first variation
In general, a first variation is just the collection of all directional derivatives
$\frac{d}{d\epsilon} \mathcal{F}(\rho+\epsilon\chi)|_{\epsilon = 0}$. For fixed $\rho$ one can treat them as a funct …
2
votes
Accepted
Is a locally invertible weak limit of injective maps injective almost everywhere?
Okay, let me try a writeup of the comment chain. For any reasonable subset $A\subset \Omega_2$ and $B := f^{-1}(A)$ you get
$$\int_A |f^{-1}(y)| dy = \int_B \det df dx \leq \liminf_{n\to\infty} \int_B …
2
votes
Accepted
Does weak continuity of Jacobians hold for non nondegenerate maps?
There is a counterexample, however there might be ways to avoid it.
Take $\mathcal{M} = \mathcal{N} =\mathbb{S}^2$, but now consider sequence of maps that cover the sphere twice, where you shrink the …
2
votes
How to interpret this quote of Lin?
This is not a full answer, since I do not know the counterexample Lin refers to, but I can offer some explanations and guesses which are too long for a comment:
You can define a first variation for cu …
3
votes
Accepted
How to interpret the vector fields $F_p(x,u,Du)$ in a Lagrangian optimization problem
There is the following interpretation coming from physics and continuum mechanics, which is a bit too long for a comment but might be helpful:
If you think of $\mathcal{F}$ as an energy that you want …