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

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 $f$ 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 $M = \mathbb{R}^3 \setminus \overline{B_1}$ where $\overline{B_1}$ is the closed unit ball. Let $g$ be an asymptotically flat metric of the form $g_{ij} = \delta_{ij}+h_{ij}$ in standard coordinat …

$\DeclareMathOperator\ddiv{div}\DeclareMathOperator\tr{tr}\newcommand{\conf}{\mathrm{conf}}$Consider this PDE on a symmetric tensor $h$ on $S^2$: $$\Delta \text{tr}(h) - \ddiv(\ddiv(h)) + \tr(h) = f$$ …

Consider the following nonlocal ODEs on $[1,\infty)$. #1) $$\begin{align} r^2 f''(r) + 2rf'(r)-l(l+1) f(r) &= -\frac{(f'(1) + f(1))}{r^2}\\ f(1) &= \alpha \\ \lim_{r\to \infty} f(r) &= 0 \end{align}$$ …

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 $u$ …

I am working on some non-local differential equations that appear in geometric analysis. One of which I posted here and was answered by @WillieWong and @losifPinelis. Consider this non-local different …

Consider this ODE on $[1, \infty)$ $(r^2 - 2ar)f''(r) + 2(r-a) f'(r) - ({4a} + m(m+1))f(r) = -4af(1) $ with initial conditions $\frac{a}{1-2a} f(1) + f'(1) = C, \qquad \lim_{r\to \infty} f(r) = 0$ whe …

Consider the following ODE: $$f'(r) = f^2(r) + O \left( \frac{1}{r^4} \right)$$ as $r$ goes to infinity. The initial conditions are $f(1) = C <0$. What is the behaviour of a solution $f$ at infinity …