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Results tagged with ap.analysis-of-pdes
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user 131004
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
3
votes
0
answers
166
views
A pde inequality
Say $M$ be a closed manifold of dimension $6$, we have
\begin{align*}
\Delta f\leq g f-f^2
\end{align*}
where $g$ is a smooth function on $M$ and $f\geq 0$ (in my case $f=|\phi|^2$ for $\phi$ a secti …
- 954
4
votes
3
answers
339
views
Reference or proof of a lemma in PDE
I am looking for a reference or proof of a lemma (if it's true) or a counter-example otherwise. It goes as follows:
Let $B_1$ and $B_2$ are two concentric balls of radius $1$ and $2$ in some $n$-dimen …
- 954
0
votes
0
answers
51
views
Coupled Kazdan-Warner type equation
Famous work of Kazdan and Warner shows that given $u\geq 0$ and a constant $c>0,$ the following equation in $f$ has a unique solution:
\begin{align*}
\Delta f+ u e^f=c
\end{align*}
I am interested in …
- 954
1
vote
1
answer
360
views
Asymptotic behaviour of solution of Kazdan-Warner equations
Let $X$ be a closed manifold. $g:X\rightarrow \mathbb{R}$ be a smooth function ,$\alpha$ a section of a line bundle with discrete zeros and $c>0$ a constant, then Kazdan-Warner's work says that the fo …
- 954
0
votes
Accepted
Asymptotic behaviour of solution of Kazdan-Warner equations
It's actually quite easy and I completely missed it. If $f$ is the solution of the original equation:
\begin{align*}
2\Delta f+\frac{e^g\lvert\alpha\lvert^2}{4}e^{5f}=c
\end{align*}
and say $f_\lambd …
- 954
1
vote
0
answers
72
views
Limiting behavior of Kazdan-Warner equations
It's a well known result by Kazdan and Warner that on a closed Riemannian manifold the pde:
\begin{align*}
\Delta f+ge^f=c
\end{align*}
has a unique solution for $g\geq 0,$ and $c$ a positive constant …
- 954
2
votes
0
answers
177
views
A $\partial\bar\partial$ type problem in Kähler Geometry
On any compact Kähler manifold $M^n$ one can ask: given a closed $p,q$ form $\alpha$ on $M$ does $\exists\beta\in\Omega^{p-1,q-1}(M;\mathbb{C})$ such that $\alpha=\partial\bar\partial\beta.$
I am inte …
- 954
2
votes
0
answers
99
views
Vortex equation on Riemann surface and a similar equation
Let's take a Riemann surface $(X,\omega)$ and a holomorphic line bundle $L$ on it with a hermitian metric $h$ on $L$. $g$ be a real valued smooth function on $X$ and we consider the following two sets …
- 954
6
votes
1
answer
193
views
Reference for $\epsilon$-regularity
I am looking for a reference for the following $\epsilon$-regularity statement: let
$(M,g)$ be a Riemannian manifold of dimension $n$,
$\Delta=dd^*+d^*d$,
$B_r$ denotes a ball of radius $r$ around a …
- 954
1
vote
1
answer
248
views
Moser iteration in dimension $6$
Let $M$ be a closed Riemannian manifold of dimension $6$. We have a function $f\geq 0$ on $M$ satisfying
\begin{align*}
\Delta f \leq gf-\frac{3}{4}f^2
\end{align*}
Where $g$ is another smooth functio …
- 954
1
vote
Accepted
Moser iteration in dimension $6$
\begin{align*}
&\Delta f\leq gf-\frac{3}{4}f^2\\
&\Rightarrow\int f^{p+1}\Delta f\leq \int(gf^{p+2}-\frac{3}{4}f^{p+3})\\
&\Rightarrow\frac{4(p+1)}{(p+2)^2}\int |\nabla(f^{\frac{p+2}{2}})| …
- 954