All Questions
7 questions
3
votes
0
answers
72
views
Compactness of bounded index solutions of the Yamabe problem
Consider, a closed Riemannian manifold $ (M^n,g) $ , $ n \geq 3 $, with positive Yamabe invariant: $$ 0< Y(M, [g]):= \inf_{0<v \in H^1} Q_g(v), $$ where $$ Q_g(v) = \inf_{0 <v \in H^1} \...
- 457
1
vote
1
answer
98
views
How to interpret the vector fields $F_p(x,u,Du)$ in a Lagrangian optimization problem
Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $C^1$ boundary. Let
$$
\begin{matrix}
F: \mathbb{R}^n \times \mathbb{R}^N \times \mathbb{R}^{nN} \to \mathbb{R},& \\
(x,z,p) \mapsto F(...
- 217
3
votes
0
answers
65
views
Existence of ground state solutions for the critical exponent
I have been recently reading Kwong's paper on the uniqueness of positive solutions for the equation $\Delta u-u+u^p=0$ in $\mathbb{R}^n$.
The authors show that the above equation has a unique positive ...
- 537
1
vote
0
answers
109
views
Is there a concentric map from the disk onto the ellipse with constant sum of singular values?
$\newcommand{Vol}{\text{Vol}}$
Let $c > 2$, and let $0<b<1$ be fixed parameters. Does there exist a $C^1$ monotone bijection $\psi:(0,1] \to (0,1]$, and a $C^1$ function $h:(0,1] \to \mathbb{...
- 6,741
4
votes
1
answer
489
views
Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism
The variational bicomplex seams to provide a modern formulation of the variational problem in terms of modern differential geometry. In particular the bigraded complex of differential forms $\Omega^{p,...
- 5,824
7
votes
2
answers
1k
views
Willmore minimizers for genus $\geq 2$
For an immersed closed surface $f: \Sigma \rightarrow \mathbb R^3$ the Willmore functional is defined as
$$
\cal W(f) = \int _{\Sigma} \frac{1}{4} |\vec H|^2 d \mu_g,
$$
where $\vec H$ is the mean ...
- 85
12
votes
1
answer
3k
views
Results about existence/uniqueness of solution to Euler-Lagrange equations?
While studying calculus of variations, there is one question that I feel is missing in the texts I'm reading:
What can we say about the existence and/or uniqueness of solutions to Euler-Lagrange ...
- 1,064