Poisson Equation: Problems & Exercises with Solutions

Consider the function $u(x,y) = x^2 - y^2$ defined on $\mathbb{R}^2$.

1. Compute $\Delta u$ and determine whether $u$ satisfies the Poisson equation $-\Delta u = f$ for some $f$. Identify $f$.
2. Now consider $v(x,y) = x^2 + y^2$. Compute $-\Delta v$ and state what equation $v$ satisfies.

Let $u(r) = r^2 \ln r$ for $r > 0$, where $r = \sqrt{x^2 + y^2}$ is the radial variable in $\mathbb{R}^2$.

1. Using the radial Laplacian $\Delta u = u''(r) + \frac{1}{r}u'(r)$, compute $\Delta u$.
2. Write the corresponding Poisson equation and identify the source term $f(r)$.

Suppose you want $u(x,y,z) = e^{x}\sin(y)\cos(z)$ to be the exact solution of a Poisson equation $-\Delta u = f$ on $\mathbb{R}^3$.

1. Compute each second-order partial derivative of $u$ and find $\Delta u$.
2. Determine the explicit source term $f(x,y,z)$ such that $u$ is the solution.

Consider the one-dimensional Poisson equation on $(0,1)$:

with $f(x) = 2$.

1. Find the general solution of $-u'' = 2$ and apply the boundary conditions to find the unique solution.
2. Verify your answer by substituting back into the equation and checking the boundary conditions.

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Suppose $u_1$ and $u_2$ are two classical solutions of: $$-\Delta u = f \text{ in } \Omega, \quad u = g \text{ on } \partial\Omega$$

1. Define $w = u_1 - u_2$. Write the boundary value problem satisfied by $w$.
2. Multiply the equation for $w$ by $w$ itself, integrate over $\Omega$, apply Green’s first identity, and conclude that $w \equiv 0$, proving uniqueness.

The fundamental solution of $-\Delta$ in $\mathbb{R}^3$ is $\Phi(x) = \dfrac{1}{4\pi |x|}$.

1. Show that $\Phi$ is harmonic (i.e., $\Delta\Phi = 0$) for $x \neq 0$ by direct computation in spherical coordinates.
2. Given a continuous, compactly supported function $f$, write the convolution formula $u = \Phi * f$ and state what Poisson equation $u$ satisfies.

Let $\Omega = \{x = (x_1,x_2,x_3)\in\mathbb{R}^3 : x_3 > 0\}$ be the upper half-space. For a source point $y = (y_1,y_2,y_3)$ with $y_3 > 0$, define the reflected point $\tilde{y} = (y_1,y_2,-y_3)$.

1. Write the Green’s function $G(x,y)$ for the Dirichlet Laplacian on $\Omega$ using the method of images, and verify that $G(x,y) = 0$ for $x_3 = 0$.
2. For $f \equiv 0$, write the Poisson integral formula giving the solution $u$ in terms of boundary data $g(x_1,x_2) = u(x_1,x_2,0)$.

Let $B = B(0,1) \subset \mathbb{R}^3$ be the open unit ball. For $y \in B$, define the Kelvin inversion $\tilde{y} = y/|y|^2$.

1. Write the Green’s function for the Dirichlet Laplacian on $B$ and verify the boundary condition $G(x,y)=0$ for $|x|=1$.
2. Using the outward normal derivative of $G$, derive the Poisson kernel $K(x,y)$ for the unit ball and write the integral formula for the harmonic extension of boundary data $g \in C(\partial B)$.

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. Consider the Neumann problem: $$-\Delta u = f \text{ in } \Omega, \quad \partial_\nu u = h \text{ on } \partial\Omega$$

1. By integrating the equation over $\Omega$ and applying the Divergence Theorem, derive the necessary compatibility condition relating $f$ and $h$.
2. Show that if $u$ is a solution, then $u + C$ is also a solution for any constant $C$, and state what additional condition makes the solution unique.

On the rectangle $\Omega=(0,L)\times(0,H)$, solve the following mixed problem using separation of variables and superposition:

with boundary conditions: $u(0,y) = 0$, $u(L,y) = 0$, $u(x,0) = 0$, and $\partial_y u(x,H) = g(x)$.

1. Seek separated solutions and derive the eigenfunctions satisfying the three homogeneous conditions.
2. Expand $g(x)$ in the appropriate eigenfunction basis and write the formal series solution $u(x,y)$.

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary, and consider: $$-\Delta u = f \text{ in } \Omega, \quad u = 0 \text{ on } \partial\Omega$$ with $f\in L^2(\Omega)$.

1. Multiply the equation by a test function $v \in H^1_0(\Omega)$, integrate over $\Omega$, and apply Green’s first identity to derive the weak formulation: find $u\in H^1_0(\Omega)$ such that $a(u,v) = F(v)$ for all $v\in H^1_0(\Omega)$. Identify the bilinear form $a(\cdot,\cdot)$ and the linear functional $F$.
2. State the Lax–Milgram theorem conditions that guarantee a unique weak solution, and verify them for this bilinear form using the Poincaré inequality.

Consider the Poisson equation with nonhomogeneous Dirichlet data: $$-\Delta u = f \text{ in } \Omega, \quad u = g \text{ on } \partial\Omega$$ where $f\in L^2(\Omega)$ and $g\in H^1(\Omega)$ (understood as a trace).

1. Introduce the lifting $\tilde{g}\in H^1(\Omega)$ with $\tilde{g}|_{\partial\Omega} = g$, define $w = u - \tilde{g}$, and write the equation satisfied by $w$ with homogeneous Dirichlet data.
2. Write the weak formulation for $w$ and invoke the Lax–Milgram theorem to conclude existence and uniqueness of the weak solution $u$.

Let $\Omega\subset\mathbb{R}^n$ be a bounded connected domain, and let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ satisfy $-\Delta u \leq 0$ in $\Omega$ (i.e., $u$ is subharmonic).

1. State the weak maximum principle for subharmonic functions and prove it by contradiction: assume the maximum is achieved at an interior point $x_0$ and derive a contradiction using the second-order condition $\Delta u(x_0) \geq 0$.
2. Use the maximum principle to prove: if $u_1, u_2\in C^2(\Omega)\cap C(\overline{\Omega})$ both solve $-\Delta u = f$ in $\Omega$ with $u_1 \leq u_2$ on $\partial\Omega$, then $u_1\leq u_2$ throughout $\overline{\Omega}$.