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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.✓
  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.✓
  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.

See pricing

50 are analysed free — try those first.

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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