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Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
3
votes
1
answer
175
views
Does the space of harmonic forms change continuously with the metric?
Let $(M,g_0)$ be a closed $n$-dimensional Riemannian manifold. Let $1<k<n$ be fixed, and let $\Delta_{g_0}:\Omega^k(M) \to \Omega^k(M)$ be the $g_0$-Laplacian. Let $H^k_{g_0}=\text{ker} \Delta_{g_0}$. …
1
vote
1
answer
287
views
Elliptic regularity of harmonic forms in $L^1$
$\newcommand{\M}{M}$
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.
…
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 …
4
votes
0
answers
168
views
Can the rank of harmonic maps decrease far from the boundary?
Let $\mathbb D^n$ be the closed unit ball in $\mathbb R^n$. Let $f:\mathbb D^n \to \mathbb{R}^n$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^n \to \mathbb{R}^n$ be t …
5
votes
1
answer
211
views
Can we perturb the Dirichlet boundary conditions to make harmonic maps locally invertible?
While analyzing a variational problem, I came to the following question:
Let $\mathbb D^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball, and let $f: \mathbb D^n \to \mathbb{R}^n$ …
5
votes
1
answer
183
views
Can harmonic maps with immersive boundary conditions have singular points?
Let $\mathbb D^2$ be the closed unit disk in $\mathbb R^2$. Let $f:\mathbb D^2 \to \mathbb{R}^2$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^2 \to \mathbb{R}^2$ be t …
1
vote
Accepted
Does the space of harmonic forms change continuously with the metric?
I think the answer is positive.
Let $D$ be the subspace of smooth closed $k$-forms on $M$. Equip $D$ with the supremum- $C^1$ norm:
$$
\| \omega \|_{C^1,sup}:=\max\{ \|\omega\|_{sup}, \|T\omega\|_{su …
0
votes
Can we specify the value of harmonic forms at a point?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\ep}{\epsilon}$
I will sketch here a different approach than the one given by Robert's answer. (It is loosely based on Alex's answer). We want to prove that …
0
votes
Accepted
Can we perturb the Dirichlet boundary conditions to make harmonic maps locally invertible?
It seems that the answer is negative for dimension $n=2$ . I am not sure if higher dimensions can be reduced to the $2D$ case.
Here is the argument for $n=2$:
Suppose that there exist $f_k \in C^{\i …
2
votes
0
answers
55
views
Approximate a one-form with nowhere vanishing one-forms having bounded Laplacians
This is a follow-up question of this one.
Let $\mathbb{D}^2$ be the closed two-dimensional unit disk (endowed with some smooth Riemannian metric), and let $\sigma \in \Omega^1(\mathbb{D}^2)$ be a sm …
4
votes
3
answers
507
views
Can we specify the value of harmonic forms at a point?
Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed.
Let $p \in M$, and let $\alpha_p \in \bigwedge^k(T_pM)^*$.
Does there exist an open neighbourhood …
3
votes
1
answer
304
views
Is this approach for establishing regularity of harmonic maps between manifolds valid?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
While trying to understand some regularity results, I thought about the following "naive" approach for establishing regularity of weakly …