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Results tagged with ap.analysis-of-pdes
Search options questions only
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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.
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
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 …
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1
vote
1
answer
248
views
Local obstructions for maps with constant singular values
$\newcommand{\M}{\mathcal{M}}$
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Let $\M, \N$ be smooth two-dimensional Riemannian manifolds.
Are there any local obstructions for the existence of a smooth map $f:\M \to …
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8
votes
0
answers
474
views
Measuring the non-commutativity of the codifferential and pullbacks
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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 $h:(0,1) \times \mathbb{S}^1 \to \mathbb{S}^1$ …
- 6,741
2
votes
0
answers
101
views
Is there a non-degenerate solution for this PDE on $\mathbb{R}^3$?
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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 …
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22
votes
0
answers
2k
views
Characterising critical points of $E(f)=\int_{M}| \bigwedge^2 df|^2 \text{Vol}_{M}$
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6
votes
0
answers
255
views
Is a Sobolev map with invertible smooth minors smooth?
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Let $k,d$ be even integers, such that $d\ge3$ and $2 \le k \le d-1$. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for som …
- 6,741
8
votes
2
answers
456
views
Obstructions for the wedge of coordinate differentials to be harmonic
Let $(M,g)$ be a smooth $d$-dimensional Riemannian manifold, $d$ even. Are there obstructions (I guess in terms of curvature) for $g$ to have the following property:
For every $p \in M$ there exist a …
- 6,741
7
votes
1
answer
208
views
Is a Sobolev map with smooth minors smooth on the whole domain?
Let $d\ge 3$ and $2 \le k \le d-1$ be integers, where at least one of $k,d$ is odd. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for some $p \ge 1$.
Q …
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13
votes
3
answers
970
views
A conformal map whose Jacobian vanishes at a point is constant?
Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$.
Assume $d \ge 3$ a …
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1
vote
0
answers
104
views
Is every "higher-order" harmonic morphism conformal?
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- 6,741
1
vote
1
answer
287
views
Elliptic regularity of harmonic forms in $L^1$
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This is a cross-post. I am looking for a reference for the regularity of harmonic forms which belong to $L^1(M)$.
Explicitly, let $\M$ be a smooth oriented Riemannian manifold.
…
- 6,741
5
votes
1
answer
160
views
Is the evaluation map from harmonic forms on the torus surjective on flat neighbourhoods?
In a nutshell:
Given a metric on the torus $\mathbb{T}^n$, can we extend any element $\sigma \in \bigwedge^k T_p^*\mathbb{T}^n$ to a global harmonic form?
Let $\mathbb{T}^n$ be the $n$-Torus. F …
- 6,741
6
votes
1
answer
388
views
Do non-continuous Sobolev maps pull back closed forms to weakly closed forms?
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