Part CCSIR NET December 2023eikonal-no-smooth-solution

Eikonal no smooth solution

Let B be the open unit disc in $R^{2}, \partial B$ its boundary and $\overset{ˉ}{B} = B \cup \partial B$ . For $λ \in (0, \infty)$ , let $S_{λ}$ be the set of twice continuously differentiable functions in B, continuous on B̄, satisfying $(u_{x})^{2} + λ (u_{y})^{2} = 1$ in B and u = 0 on $\partial B$ . Which of the following statements are true?

A. $S_{1} = \emptyset$ ✓
B. $S_{2} = \emptyset$ ✓
C. $S_{1}$ has exactly one element and $S_{2}$ has exactly two elements.
D. $S_{1}$ and $S_{2}$ are both infinite.

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

Eikonal no smooth solution

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