Notes › MATH 5339: Partial Differential Equations Lecture 6
Harmonic Functions
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Table of Contents
Harmonic Functions
- Any function whose Laplacian sum to zero
- For any simple region that is connected, open, and bounded, then you will know the behavior of the function of the inside if you know the behavior along the boundary (Dirichlet problem)
- Similarly for the derivative of the function along the boundary (Neumann) or the derivative plus the original function along the boundary
Max/Min Principle
Definition 06.1 (Max/Min Principle).
Let $D$ be a connected bounded open set. Let $u$ be a harmonic function. Then, $u$ attains its maximum and minimum values on the boundary of $D$.
Proposition 06.2 (Two harmonic functions).
If $u$ and $v$ solve
$$\begin{cases} \delta u = f, &(x, y) \in D \\ u |_{\nabla D} = h \end{cases}$$then $u=v$ on $D$
6.2 Mixed Boundary Conditions
Example
