--- title: "Laplace's Equation and the Maximum Principle" description: "A solution of Laplace's equation equals its own average over any small sphere, so it can have no interior peak. Uniqueness and stability for the boundary value problem follow from that one observation" canonical: https://lightmysky.com/learn/mathematics/laplaces-equation-and-the-maximum-principle-mt_jSEkaTav1V source: https://lightmysky.com/learn/mathematics/laplaces-equation-and-the-maximum-principle-mt_jSEkaTav1V.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Laplace's Equation and the Maximum Principle A solution of Laplace's equation equals its own average over any small sphere, so it can have no interior peak. Uniqueness and stability for the boundary value problem follow from that one observation. Subject: Mathematics · Area: Differential Equations · Ages 22 to 23 Page: https://lightmysky.com/learn/mathematics/laplaces-equation-and-the-maximum-principle-mt_jSEkaTav1V ## Ready when they can - State the mean value property and derive the maximum principle from it - Use the maximum principle to prove uniqueness for the Dirichlet problem - Explain why harmonic functions are smooth even when the boundary data is rough ## Lesson: Steady landscapes with no interior peaks A harmonic function solves Laplace's equation u_xx plus u_yy equals 0. Every such function has the mean value property: its average over any circle whose disc lies inside the region equals its value at the center, whatever the radius. For example, u(x,y) equals 3x plus 4 is harmonic, and its average over any circle centered at (1,1) is just u(1,1), which is 7. Not every formula qualifies: x squared plus y squared has Laplacian 4, while x squared minus y squared has Laplacian 0. **Example.** Take u(x,y) equals x squared minus y squared and a small circle centered at (2,1). Since u_xx is 2 and u_yy is negative 2, the function is harmonic. The mean value property says the circle average equals the center value u(2,1), which is 4 minus 1, giving 3. For a linear u on a closed disc, the maximum sits on the rim in the direction of steepest rise: on the closed unit disc about the origin, 3x plus 4y peaks at 5, the length of (3, 4). From averaging comes the maximum principle: on a bounded region, a nonconstant harmonic function that is continuous up to the boundary attains its maximum and minimum only on the boundary, never inside. Uniqueness for the Dirichlet problem follows. If two such solutions share the same boundary data, their difference is harmonic with zero boundary values, so its max and min are both zero and the solutions agree everywhere. Boundary values between 0 and 5 likewise trap the whole interior between 0 and 5. Harmonic functions are smooth at every interior point, even when the boundary data is rough or jumps. Averaging is what smooths: each interior value blends the whole boundary, so no interior point inherits a jump or a corner. At the centre of a disc the value is the plain average of the boundary values. Think of electrostatics: with no interior charge, the boundary voltages fix one gentle landscape inside. **Recap.** Averages equal center values, so peaks live on the boundary and boundary data fixes the interior. ## Practice 19 questions on this page, each with its working shown. ## Needs first - [The Wave Equation and Its Characteristics](https://lightmysky.com/learn/mathematics/the-wave-equation-and-its-characteristics-mt_-_QCiBWg1w) ## Optional connections - [Divergence and Curl](https://lightmysky.com/learn/mathematics/divergence-and-curl-mt_JbjW8WKOPE) ## Opens up - [Weak Solutions and Test Functions](https://lightmysky.com/learn/mathematics/weak-solutions-and-test-functions-mt_AKlC3Ol70p) - [Harmonic Functions and What Analyticity Forces](https://lightmysky.com/learn/mathematics/harmonic-functions-and-what-analyticity-forces-mt_jqajP39XI4)