Uniqueness theorem for Poisson's equation: Difference between revisions

Revision as of 17:24, 6 January 2021

Revision as of 17:25, 6 January 2021

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:<math>\int_{S} (\varphi \, \mathbf{\nabla}\varphi) \cdot \mathrm{d}\mathbf{S}= \int_V (\mathbf{\nabla}\varphi)^2 \, \mathrm{d}V. \qquad (2)</math>

If the Dirichlet boundary condition is satisfied on $<math>S$</math> by both solutions (i.e., if <math>\varphi = 0</math> on the boundary), then the left-hand side of <math>(2)</math> is zero. Consequently, we find that

:<math>\int_V (\mathbf{\nabla}\varphi)^2 \, \mathrm{d}V = 0.</math>

However, because this is the volume integral of a positive quantity (due to the squared term), we must have