All Questions
Tagged with dg.differential-geometry elliptic-pde
64 questions with no upvoted or accepted answers
1
vote
0
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110
views
Moser iteration epsilon-regularity for non-linear system in general dimension
I am attempting to prove the following result in general dimension $n$. Given $(M^n,g)$ a Riemannian manifold with $\mathrm{Ric}_g \geq -(n-1)$ and $\mathrm{Vol}_g(B_1(x)) \geq v > 0$ for all $x \...
1
vote
0
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73
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
1
vote
0
answers
98
views
The module generated by kernel of an elliptic differential operator
Let $D$ be an elliptic differential operator defined on $\Gamma(E)$ where $\Gamma (E)$ is the space of smooth sections of vector bundle $E$ over a smooth manifold $M$. So $\Gamma (E)$ is a $C^\...
- 356
1
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0
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235
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Fredholmness of elliptic operator on Hölder spaces
Let $(M,g)$ be a smooth oriented closed Riemannian manifold, $E\to M$ a smooth vector bundle, and $C^{k,\alpha}(E)$ the Banach space of sections of $E$ that are $k$-times differentiable (with respect ...
- 163
1
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0
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52
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Laplacian eigenvalue problem for systems coupled along the boundary
I am looking for references on eigenvalue problems for systems of the following type:
Let $\Omega$ be the region enclosed by a right triangle with legs $\Gamma_1$, $\Gamma_2$, and hypotenuse $\...
1
vote
0
answers
54
views
Geometric interpretation for uniformly elliptic pde of 2 second order
Let $\Omega \subset \mathbb{R}^{2}$ a domain,let $u \in C^{2}(\Omega)$, the operator
$Lu= tr(A.D^{2}u) + <\nabla u,b> +cu$
where $A$ is a symmetric matrix, $b$ is a vector field continuous ...
- 381
1
vote
0
answers
76
views
PDE on an open ball with prescribed value on some open subsets
Suppose that we have a $r$-order differential operator $L:C^r(B^n, \mathbb{R})\to C^0(B^n,\mathbb{R})$ where $B^n $ is the open unitary ball in $\mathbb{R}^n$ (we can assume for simplicity that it ...
- 2,533
1
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0
answers
117
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Degenerate Monge-Ampere equation on a bounded domain with $C^{2,1}$ boundary
In the paper by Guan Pengfei: "C^2 a priori estimates for degenerate Monge-Ampere equations" https://projecteuclid.org/euclid.dmj/1077242669
Prof. P. Guan proved in Theorem 1 that the degenerate Monge-...
- 81
1
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0
answers
250
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A question about the proof of McLean's theorem: why is the space of $C^{k,\alpha}$ exact $p$-forms a Banach space?
In the famous paper "Deformations of Calibrated Submanifolds" by Robert McLean, he showed that given a smooth compact special Lagrangian $L$ in a Calabi-Yau manifold $(X^{2n},\omega,\Omega)$, there is ...
- 461
1
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0
answers
113
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upper bound for heat kernel of Grushin operator
Let $\Omega$ be a bounded open domain in $\mathbb{R}^{n+1}$ with smooth boundary. $\Omega\cap\{(0,\cdots,0,y)\in\mathbb{R}^{n+1}| y\in \mathbb{R}\}\neq \varnothing$
Consider the sub-elliptic operator
...
- 287
0
votes
0
answers
106
views
Using a theorem (which is originally set on 2-dim bounded domain in Euclidean space) on a torus
Actually I'm reading a paper on mean-field equation on torus by M.Struwe and G.Tarantello Here, they studied $$\tag{1}
-\Delta u=\lambda\left(\frac{e^u}{\int_{\Omega} e^u d x}-\frac{1}{|\Omega|}\right)...
- 809
0
votes
0
answers
135
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Relative bounds for vorticity
Write the vorticity equation as
\begin{equation}\label{Eq20}
\begin{split}
\dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...
- 317
0
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0
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146
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Generalized harmonic map
Let $M, N$ be closed Riemannian manifolds and $c$ be a constant. For a map $f:M\to N$, define the energy as
$$E(f) = \frac{1}{2} \int_M\Big( \| df(x)\|^2 - c\| f(x) \|^2 \Big) d\mu(x).$$
When $c=0$, ...
- 169
0
votes
0
answers
158
views
positive eigenfunction on complete Riemannian manifold
Let $(M^n,g)$ be a complete(non-compact) Riemannian manifold. Consider the positive solution to the equation
$$
\Delta u = u
$$
where $\Delta=\nabla_i \nabla_i$ is negative semi-definite. Is there ...
- 98