# Hence for each choice of m and n we obtain a product

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Unformatted text preview: lace’s equation, we need boundary conditions which will impose inﬁnitely many constraints. To see what boundary conditions are natural to impose, we need to think of a physical problem which leads to Laplace’s equation. Suppose that u(x, y ) represents the steady-state distribution of temperature throughout a uniform slab in the shape of a region D in the (x, y )-plane. If we specify the temperature on the boundary of the region, say by setting up heaters and refrigerators controlled by thermostats along the boundary, we might expect that the temperature inside the room would be uniquely determined. We need inﬁnitely many heaters and refrigerators because there are inﬁnitely many points on the boundary. Specifying the temperature at each point of the boundary imposes inﬁnitely many constraints on the harmonic function which realizes the steady-state temperature within D. The Dirichlet Problem for Laplace’s Equation. Let D be a bounded region in the (x, y )-plane which is bounded by a curve ∂D, a...
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## This document was uploaded on 01/12/2014.

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