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Part CCSIR NET December 2025the-maximum-principle-caps-u-by-the-boundary-datas-own-range-e-to-plus-or-minus-root-2-nothing-outside-it-is-ever-attained

The maximum principle caps u by the boundary datas own range e to plus or minus root 2 nothing outside it is ever attained

Consider the boundary value problem (BVP)uxx+uyy=0) u_{xx}+u_yy=0 in Ω=\Omega={(x,y)∈R2:x2+y2<1(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}<1}, u(x,y)=e^(x+y) on ∂Ω=\partial\Omega={(x,y)∈R2:x2+y2=1(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}=1}. Then which of the following statements are true?

  1. A.There exists a unique solution to BVP.✓ Correct
  2. B.The BVP does NOT have a solution.
  3. C.There exists a solution u to BVP such that u(x,y)=(1+e)/2 for some (x,y)∈Ω∪∂Ω(x,y)\in\Omega\cup\partial\Omega.✓ Correct
  4. D.There exists a solution u to BVP such that u(x,y)=(1+e3)/2u(x,y)=(1+e^{3})/2 for some (x,y)∈Ω∪∂Ω(x,y)\in\Omega\cup\partial\Omega.

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 boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

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

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