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 \((0,1)\):

\[ -u”(x) = f(x), \quad x \in (0,1), \quad u(0) = 0,\; u(1) = 0 \]

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:

\[ -\Delta u = 0 \text{ in } \Omega \]

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 \(\overline{\Omega}\).