Partial Differential Equations: An Introduction (Dover Books on Mathematics) by Colton David

Author:Colton, David [Colton, David]
Language: eng
Format: azw3
ISBN: 9780486138435
Publisher: Dover Publications
Published: 2012-06-13T16:00:00+00:00

4.3 BOUNDARY VALUE PROBLEMS FOR LAPLACE’S EQUATION

In this section we formulate the classical boundary value problems for Laplace’s equation. We do not, however, consider initial-value problems, since for Laplace’s equation they are improperly posed. To see this, consider the Cauchy problem

The (unique) solution of this problem is

But as n tends to infinity the initial data tends to zero whereas the solution oscillates between arbitrarily large values, and the solution does not depend continuously on the initial data. Although the Cauchy problem for Laplace’s equation is improperly posed, it does appear in certain situations of practical importance. For details of how to “solve” such a problem we refer the reader to Payne.

The problems which are well posed for Laplace’s equation are boundary-value problems which we shall now formulate. In what follows, D is a bounded, simply connected domain in R2 with C2 boundary ∂D, and f is a continuous function defined on ∂D. The unit outward normal to ∂D will again be denoted by v. The following boundary-value problems are the classical ones associated with Laplace’s equation.

Interior Dirichlet Problem: Find a function u ∈ C2 (D) ∩ () such that u(x) is harmonic in D and u(x) = f(x) for x on ∂D.

Exterior Dirichlet Problem: Find a function u E C2(R2\5) ∩ C(R2\D) such that u(x) is harmonic in R2\, bounded in R2\D, and u(x) = f(x) for x on ∂D.

Interior Neumann Problem: Find a function u E C2(D) ∩ C1() such that u(x) is harmonic in D and ∂u(x)/∂v = f(x) for x on ∂D. Note that from (4.4) we must require f(x) to satisfy

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