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Consider the manifold $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball. Equip $M$ with an asymptotically flat metric $g$ of high order. Let $\gamma$ be the induced metric on $\partial M$ …

Let $M = \mathbb R^3 \setminus B_1$ where $B_1$ is the unit ball. Let $ h \in C^{\infty}(\partial M)$ and let $u_0$ be the unique function that vanishes at infinity and solves $$\begin{cases} -\Delta …

This is a similar question to this but with a different ODE. Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer, $R>1$, …

Let $M = [1,\infty) \times S^2$. Consider the weighted Sobolev space $H^k_{\delta}(M)$ with the Sobolev norm: $$\lVert u \rVert_{k,\delta}^2 := \sum_{n=0}^k \int_M |D^nu \,r^{n-\delta}|^2 r^{-3} dV $$ …

The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm $$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$ The space $L^2([a,b]\times S^2)$ i …

Let $M = [1,\infty)\times S^2$. For an integer $k \geq 2$ and number $\tau<0$, define the space $L^2_{\tau}([1,\infty);H^k(S^2))$ to be all $H^k(S^2)$-valued functions $u$ on $[1,\infty)$ with $\lVert …

Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer and $a$ be a real number. Let $u(r)$ be a function on $[1,\infty)$ so …

Let $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball. It is known that for every $g \in H^{\frac{1}{2}} (\partial M)$, and for an appropriate $\delta$, there exists a unique solution $u$ …

Let $M=[0,\infty) \times S^2$. We have the regular regular Sobolev space $H^1(M)$. We also have the space $H^1\bigg([0,\infty); H^1(S^2)\bigg)$. Are those two spaces the same? Does one contain the oth …

Let $g$ be an asymptotically flat metric on $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball. Suppose $X$ is a smooth vector field on $M$ that is decaying exponentially and satisfies $$\D …

Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer. Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that …

Let $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is unit ball. I am trying to solve the following PDE for $f$: $$\Delta f -\frac{ f }{r^2}+ \frac{ \left. f \right|_{\partial M}}{r^2} = 0, \qquad \text …

The needed bounds on the Legendre functions $P_{\ell}$, $Q_{\ell}$ are proven here: https://arxiv.org/abs/2411.02801 See appendix A.3