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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 …

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 …

Consider the PDE $$\Delta f + \lambda f = g$$ on $S^2$, where $\Delta$ is with respect to the round metric, $g \in L^2(S^2)$ and $\lambda>0$. I wish to study the existence and uniqueness of this PDE f …

Suppose for simplicity that $M$ is a compact n-dimensional submanifold of $\mathbb{R}^n$ with boundary. Extend $f$ to all $\mathbb{R}^n$ so that it's $0$ outside $M$ and define the function $u_1(x) := …

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 $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball (equip $M$ with the euclidean metric for simplicity, but it will be replaced by an arbitrary asymptotically flat metric). Let $N: L^2( …

Consider the manifold $\mathbb{R}^3 \setminus B$ where $B$ is the ball with radius 1 with riemannian metric $g$ (not necessarily the euclidean metric). I am looking for solutions to $\Delta_g f = 0$ …

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 $u$ …