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Results tagged with ap.analysis-of-pdes
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user 46290
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
4
votes
1
answer
273
views
Can every diffeomorphism be rescaled into a volume preserving one?
This is a cross-post. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk, and let $f:D \to D$ be a diffeomorphism.
Does there exist a smooth $h \in C^{\infty}(D)$ such that $h\cdot f$ is an ar …
- 45k
9
votes
0
answers
287
views
Is there a variational interpretation for the equation $\operatorname{div}(\star \circ \bigw...
$\newcommand{\id}{\operatorname{Id}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\TM}{\operatorname{TM}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Cof}{\operatorname{Cof}}
\newcommand{\Det}{\oper …
- 6,741
2
votes
0
answers
101
views
Is there a non-degenerate solution for this PDE on $\mathbb{R}^3$?
$\newcommand{\tr}{\operatorname{tr}}$
$\newcommand{\R}{\mathbb{R}}$
Does there exist a smooth map $f:\mathbb{R}^3 \to \mathbb{R}^3$, which satisfies
$$\tr \big( df \otimes \delta(df \wedge df) \big)=0 …
- 6,741
1
vote
0
answers
102
views
A PDE involving a diffeomorphism of $\mathbb{S}^1$
This question is a special case of this one.
Let $s(\theta)>0, b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$.
Do there exist a diffeomorphism $\phi:\mathbb{S}^1 \to \ma …
- 6,741
2
votes
0
answers
143
views
Does this geometric PDE have a solution?
Let $s(\theta), b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$, and assume that $s$ never vanishes.
Does there exist a map $h:(0,1) \times \mathbb{S}^1 \to \mathbb{S}^1$ …
- 26.3k
5
votes
1
answer
595
views
A vector field whose flow has constant singular values
$\newcommand{\tr}{\operatorname{tr}}$
$\renewcommand{\div}{\operatorname{div}}$
Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Given a vector field $X$ on $D$, let $\psi_t$ be its flow.
Does …
- 108k
2
votes
1
answer
164
views
The only rotation fields satisfying this PDE are constant
$\newcommand{\div}{\operatorname{div}}$$\newcommand{\SO}{\operatorname{SO(2)}}$$\newcommand{\R}{\operatorname{\mathbb{R}}}$$\newcommand{\bdx}{\partial_x}$$\newcommand{\bdy}{\partial_y}$$\newcommand{\t …
- 39k
1
vote
2
answers
309
views
Is there an area-preserving concentric diffeomorphism of the ellipse?
$\DeclareMathOperator\Vol{Vol}$This is a cross-post.
Let $0<b<1$ be a fixed parameter, and let $(R(\theta),\theta)$ be the polar coordinates of the ellipse
$$E=\{(x,y) \in \mathbb R^2 \, | \, \Bigl(\ …
- 619
1
vote
0
answers
108
views
Is there a concentric map from the disk onto the ellipse with constant sum of singular values?
$\newcommand{Vol}{\text{Vol}}$
Let $c > 2$, and let $0<b<1$ be fixed parameters. Does there exist a $C^1$ monotone bijection $\psi:(0,1] \to (0,1]$, and a $C^1$ function $h:(0,1] \to \mathbb{R}$ that …
- 6,741
3
votes
0
answers
112
views
Are continuous harmonic maps between Riemannian manifolds smooth up to the boundary?
Let $M,N$ be smooth, connected, compact, oriented, two-dimensional Riemannian manifolds, with $C^k$ boundaries.
Let $f:M \to N$ be a Lipschitz continuous weakly harmonic map**, and assume that $f(\par …
- 6,741
1
vote
1
answer
248
views
Local obstructions for maps with constant singular values
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M, \N$ be smooth two-dimensional Riemannian manifolds.
Are there any local obstructions for the existence of a smooth map $f:\M \to …
- 108k
0
votes
0
answers
152
views
Metric obstructions for area-preserving diffeomorphisms with constant singular values
Let $\mathbb{T}^2$ be the topological $2$-dimensional torus, and let $0<\sigma_1 < \sigma_2$ satisfy $\sigma_1 \sigma_2=1$. Let $g$ be an arbitrary smooth Riemannian metric on $\mathbb{T}^2$.
Does the …
- 6,741
7
votes
1
answer
403
views
A diffeomorphism of the torus with constant singular values
Let $\mathbb{T}^2=\mathbb{S}^1 \times \mathbb{S}^1$ be the flat $2$-dimensional torus, and let $0<\sigma_1 < \sigma_2$ satisfy $\sigma_1 \sigma_2=1$.
Does there exist an area-preserving diffeomorphism …
- 6,741
2
votes
3
answers
258
views
How to show continuity and monotonicity of solutions to this parametrized equation?
Let $1 \le p <2$ be a parameter. Consider the equation
$$
\frac{2^{p/2} (1-\sqrt{s})^p-1}{\sqrt{s}}=-2^{p/2-1}p(1-\sqrt{s})^{p-1}. \tag{1}
$$
I am rather certain that for each $1 \le p <2$, there is u …
- 60.5k
2
votes
1
answer
372
views
Is there a diffeomorphism of the disk with constant sum of singular values?
This question is a relaxed version of this question.
Let $D \subseteq \mathbb{R}^2$ be the closed unit disk, and let $c \ge 2$.
Does there exist a diffeomorphism $f:D \to D$ with constant sum of …
- 6,741