# Tagged Questions

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### 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 $Q$. ...
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### 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 ...
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### 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 ...
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### 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 $\Omega_2(t)$. ...
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### 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 ...
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### $C^1$ 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 ...
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 $$\int_\Gamma ... 1answer 26 views ### Lipschitz domain and surface measure Let S be the boundary of a Lipschitz domain \Omega. We know it has a surface measure \mu. Can we write d\mu = f(x)dx with f explicity given in terms of the Lipschitz maps that make up the ... 1answer 72 views ### Why is partition of unity required in definition of Sobolev space on manfolds? Why do we need to use \phi_i u in the expression for the norm? Why not just u? The range of integration is over R(x_i) anyway, so I don't understand why it is necessary. If you check Kendall, ... 0answers 63 views ### An imcomprehensible proof on Arnold's Lectures on PDE, contact diffeomorphism In Lectures on Partial Differential Equations, Arnold gives a theorem in the chapter Huygens' Principle in the Theory of Wave Propagation: Theorem 1 (the theory of support functions). The manifold ... 1answer 72 views ### Construct harmonic function on noncompact manifold M is a non-compact Riemannian manifold, p \in M. Consider Dirichlet problems: \Delta u = 0 in {B_p}\left( i \right) (i = 1,2, \dots ), u{|_{\partial {B_p}\left( i \right)}} = {f_i}, {f_i} ... 0answers 100 views ### Sobolev trace theorem for manifolds with boundary Can I get a reference to a trace theorem for a manifold M with boundary \partial M, and I am hoping the inequality$$\lVert Tu \rVert_{L^2(\partial M)} \leq c\lVert u \rVert_{H^1(M)}$$will hold. ... 1answer 173 views ### Green's function for the Yamabe problem I'm currently reading the paper on the Yamabe problem by Lee and Parker, and am looking for a reference for Theorem 2.8. Theorem 2.8 (Existence of the Green Function). Suppose M is a ... 1answer 180 views ### Submersion Theorem for Banach Spaces I'm having difficulty proving a well-known result from functional analysis. Any hints would be greatly appreciated. Fix a Fréchet differentiable map of Banach spaces g: X \to B. Assume that, at a ... 2answers 766 views ### PDEs on manifold: what changes from Euclidean case? I know some PDE theory for nice open domains in \mathbb{R}^n. I want to know what the changes are when I switch to other domains like manifolds. For example, do things like Poincare's inequality ... 1answer 55 views ### meaning of “doubly inward-pointing” I am currently trying to understand the b-calculus developed by R. Melrose. an important part of the theory is the stretched product of a manifold X with boundary \partial X. looking at the ... 2answers 350 views ### Green's Function for Operator I'm trying to show that the Green's function for the Laplace operator -\nabla^2 is badly behaved at infinity. I.e.$$\int d^dx|G(x,y)|^2=\infty for $d=1,2,3$. What happens when $d>4$? I know ...
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 ...