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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.
- Winter '14