Uniqueness of Solution

Poisson's equation for $ has a unique solution given an appropriate specification of the boundary conditions across the entire surface, including: 1) the potential $, or 2) its normal derivative d$/dn, is specified on the boundary (Jackson 1975). These are called Dirichlet and Neumann boundary conditions, respectively. Mixed boundary conditions are also possible, in which $ and d$/dn are are specified on non-overlapping parts of the boundary. (Specifying both $ and d$/dn over any part of the boundary is an overspecifica-tion of the problem, and the existence of a solution is not guaranteed.) This uniqueness property allows us to be creative in how we derive the solution, since finding any solution to Poisson's equation which matches the boundary conditions implies that we have found the solution.

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