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Reading a bit about the Laplace equation in some lecture notes, appeared the following questions:

(1) What would a Cauchy problem for the Laplace equation?

(2) What does it mean to be a problem well put?

Thanks!

liamdalton
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Anderson Lima
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1 Answers1

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A 'Cauchy problem' is one with values of the solution (or its derivatives, or some combination) specified on a hypersurface (a set one dimension smaller than the problem domain). So, if you're solving solving Laplace's eqn. on some open, connected set $\Omega$, then the Cauchy problem could be specifying $u$ on $\partial \Omega$, and then solving $\Delta u = 0$ on $\Omega$ subject to that restriction.

For example, $$ \begin{cases} \Delta u = 0; \quad |x| < 1\\ u = f(x); \quad |x| = 1 \end{cases} $$

is a Cauchy problem on the unit disc. (alternatively, we could have specified the normal derivative of u at $|x| = 1$, or other sorts of conditions)

A problem is 'well-posed' if

  1. A solution exists
  2. The solution is unique
  3. The solution depends continuously on the data (in this case the boundary values)

The last condition just means, if you change the boundary data a little bit, the solution only changes a little bit.

BaronVT
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