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Part BCSIR NET December 2024the-exterior-problem-needs-a-condition-at-infinity

The exterior problem needs a condition at infinity

The problem ∂2u/∂x2+∂2u/∂y2=0\partial^{2}u/\partial{}x^{2} + \partial^{2}u/\partial{}y^{2} = 0 in {(x,y)∈R2:x2+y2>1(x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} > 1}, u(x, y) = 1 on {(x,y)∈R2:x2+y2=1(x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} = 1} has

  1. A.no solution
  2. B.exactly one solution
  3. C.exactly two solutions
  4. D.infinitely many solutions✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests existence vs uniqueness.

See pricing

50 are analysed free — try those first.

The trap it tests

Existence vs uniqueness

A theorem giving one was read as giving both.

Drill statements like this

Related counterexample: All the classical equations satisfy a maximum principle

More on this topic

The chapter behind this: The three classical equations — free to read

From Partial Differential Equations › Laplace, heat and wave equations: separation of variables

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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