How can we show that a Dirichlet problem for Laplace's equation in a finite region has a unique solution.
Usually we can consider u2 - u1, a difference in values.
2026-03-27 17:40:35.1774633235
dirichlet problem for laplace's equation
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If $u_1$ and $u_2$ solve Laplace's equation on the same domain with the same boundary conditions, then $u_2 - u_1$ solves Laplace's equation with $0$ boundary conditions. The maximum principle now implies that $u_2 - u_1 = 0$.