1
vote
1answer
42 views

Definition of fractional Laplacian on a compact manifold?

How does one define the fractional Laplacian operator $(-\Delta)^s$ on a compact Riemannian manifold? In $\mathbb{R}^n$, it is defined $$ (-\Delta)^s f(x) = c_{n,s} \int_{\mathbb{R}^n} \frac{f(x) - ...
3
votes
0answers
36 views

Variational Problem on a manifold - bounding my functional from below

Let $(M, g_{ij})$ be a compact Riemannian manifold. Let $f \in C^{0,\alpha}(M)$ be a given function. I have the linear operator $L = \Delta h - 12u^2h$, where $u \in C^{2,\alpha}$. I need to show ...
1
vote
1answer
26 views

Transferring a result in PDE from open domain to a manifold

Can anyone recommend me something that in detail, talks about transferring a result in Sobolev space (as opposed to Holder spaces or something like that) that holds for open domains in $\mathbb{R}^n$ ...
1
vote
1answer
35 views

Is $\lVert u \rVert_{L^2(M)} + \lVert \Delta u \rVert_{L^2(M)}$ equivalent to $\lVert u \rVert_{H^2(M)}$?

On a bounded Riemannian manifold without boundary, is it true that the norms $$\lVert u \rVert_{L^2(M)} + \lVert \Delta u \rVert_{L^2(M)}$$ is equivalent to the full $H^2$ norm $\lVert u ...
2
votes
1answer
85 views

Why should the diffusivity matrix (of elliptic operator) map tangent space to itself?

I have seen that an elliptic operator $A$ on a hypersurface $\Gamma$, written as $$Au=-\nabla_\Gamma \cdot (M(x)\nabla_\Gamma u)$$ (where $\nabla_\Gamma$ is the tangential or surface gradient) is ...
1
vote
2answers
78 views

Distributional derivatives on hypersurface?

In a paper I was reading, the define a set $Q=(0,T)\times \Omega$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain, and then they write $$\langle \frac{d}{dt}u - \Delta u, \varphi ...
0
votes
0answers
26 views

About a function space on $\bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}$

For each $t$, let $\Gamma(t)$ be a $C^k$ hypersurface without boundary. Define $$Q = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$ I am trying to understand some properties of this space $Q$. ...
1
vote
0answers
25 views

Don't understand an integration by parts result involving a step function on spacetime domain

I'm reading this work. Let $\Omega$ be a bounded (open) domain, and define $Q=(0,T)\times\Omega$. For every $t \in [0,T]$, let $\Omega_1(t), \Omega_2(t)$ be open subsets of $\Omega$, with $S(t)$ the ...
4
votes
1answer
78 views

How to show the identity $\int_0^T \int_{\Gamma(t)}f(s,t)\;dsdt = \int_S f(\sigma)(1+(\mathbf w \cdot \mathbf n)^2)^{-\frac{1}{2}}\;d\sigma$?

I am reading this paper. Let $\Gamma(t)$ be a smooth closed connected oriented hypersurface for each $t \in [0,T]$. Define the set $$S = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$ On page 5 of ...
0
votes
1answer
18 views

About the boundary of a set of the form $Q_i = \bigcup_{t \in (0,T)}\Omega_i(t) \times \{t\}$

Let $\Omega$ be a bounded (open) domain. For every $t \in [0,T]$, let $\Omega_1(t), \Omega_2(t)$ be open subsets of $\Omega$, with $S(t)$ the interface separating $\Omega_1(t)$ and $\Omega_2(t)$. ...
6
votes
0answers
34 views

Elliptic Regularity on Manifolds

Sorry if this has been asked before. Are there are books which talk about elliptic regularity on manifolds? Everything I've been able to find has been about $\mathbb{R}^N$. Can every question about ...
0
votes
1answer
55 views

$C^1$ domains vs. Lipschitz domains in PDEs, do we need transformation of coordinates?

I am getting a bit confused. In the definition of $C^1$ manifold in Renardy and Rogers, they say that $\partial\Omega$ is of class $C^1$ if every point on $\partial\Omega$ has a neighbourhood within ...
1
vote
1answer
36 views

Bounding an integral on boundary of Lipschitz domain

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain with bounded boundary $\Gamma.$ So $\Gamma$ is a hypersurface of dimension $(n-1)$. I want to show that $$\int_\Gamma ...
0
votes
1answer
26 views

Lipschitz domain and surface measure

Let $S$ be the boundary of a Lipschitz domain $\Omega$. We know it has a surface measure $\mu$. Can we write $d\mu = f(x)dx$ with $f$ explicity given in terms of the Lipschitz maps that make up the ...
2
votes
1answer
72 views

Why is partition of unity required in definition of Sobolev space on manfolds?

Why do we need to use $\phi_i u$ in the expression for the norm? Why not just $u$? The range of integration is over $R(x_i)$ anyway, so I don't understand why it is necessary. If you check Kendall, ...
4
votes
0answers
63 views

An imcomprehensible proof on Arnold's Lectures on PDE, contact diffeomorphism

In Lectures on Partial Differential Equations, Arnold gives a theorem in the chapter Huygens' Principle in the Theory of Wave Propagation: Theorem 1 (the theory of support functions). The manifold ...
3
votes
1answer
72 views

Construct harmonic function on noncompact manifold

$M$ is a non-compact Riemannian manifold, $p \in M$. Consider Dirichlet problems: $\Delta u = 0$ in ${B_p}\left( i \right)$ ($i = 1,2, \dots $), $u{|_{\partial {B_p}\left( i \right)}} = {f_i}$, ${f_i} ...
1
vote
0answers
100 views

Sobolev trace theorem for manifolds with boundary

Can I get a reference to a trace theorem for a manifold $M$ with boundary $\partial M$, and I am hoping the inequality $$\lVert Tu \rVert_{L^2(\partial M)} \leq c\lVert u \rVert_{H^1(M)}$$ will hold. ...
4
votes
1answer
173 views

Green's function for the Yamabe problem

I'm currently reading the paper on the Yamabe problem by Lee and Parker, and am looking for a reference for Theorem 2.8. Theorem 2.8 (Existence of the Green Function). Suppose $M$ is a ...
3
votes
1answer
180 views

Submersion Theorem for Banach Spaces

I'm having difficulty proving a well-known result from functional analysis. Any hints would be greatly appreciated. Fix a Fréchet differentiable map of Banach spaces $g: X \to B$. Assume that, at a ...
25
votes
2answers
766 views

PDEs on manifold: what changes from Euclidean case?

I know some PDE theory for nice open domains in $\mathbb{R}^n.$ I want to know what the changes are when I switch to other domains like manifolds. For example, do things like Poincare's inequality ...
0
votes
1answer
55 views

meaning of “doubly inward-pointing”

I am currently trying to understand the $b$-calculus developed by R. Melrose. an important part of the theory is the stretched product of a manifold $X$ with boundary $\partial X$. looking at the ...
3
votes
2answers
350 views

Green's Function for Operator

I'm trying to show that the Green's function for the Laplace operator $-\nabla^2$ is badly behaved at infinity. I.e. $$\int d^dx|G(x,y)|^2=\infty$$ for $d=1,2,3$. What happens when $d>4$? I know ...
3
votes
2answers
268 views

Does a diffeomorphism between manifolds induce an isomorphism of Sobolev spaces?

Let $M$ be a Riemannian manifold, and define the Sobolev spaces $H^k(M)$ to be the set of distributions $f$ on $M$ such that $Pf \in L^2(M)$ for all differential operators $P$ on $M$ of order less ...