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MMLU-Pro / 8418 / Suppose Ω is a bounded open area in R^(n). For any …

Problem

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question

Suppose Ω\Omega is a bounded open area in Rn\mathbb{R}^n. For any f∈L2(Ω)f\in L^2(\Omega), the Laplace equation (with respect to a real function uu), Δu=f\Delta u = f with boundary condition u∣∂Ω=0u\mid_{\partial \Omega}=0, has a unique weak solution. This can be proved by: 1. Poincare inequality and Riesz representation theorem; 2. Cauchy-Schwartz inequality and Hahn-Banach theorem. 3. None of the above. Return the answer as a number
Plain-text mathematical notation (without MathML)
Suppose Ω is a bounded open area in R^(n). For any f∈L²(Ω), the Laplace equation (with respect to a real function u), Δu=f with boundary condition u∣_(∂Ω)=0, has a unique weak solution. This can be proved by: 1. Poincare inequality and Riesz representation theorem; 2. Cauchy-Schwartz inequality and Hahn-Banach theorem. 3. None of the above. Return the answer as a number
Original LaTeX notation
Suppose $\Omega$ is a bounded open area in $\mathbb{R}^n$. For any $f\in L^2(\Omega)$, the Laplace equation (with respect to a real function $u$), $\Delta u = f$ with boundary condition $u\mid_{\partial \Omega}=0$, has a unique weak solution. This can be proved by: 1. Poincare inequality and Riesz representation theorem; 2. Cauchy-Schwartz inequality and Hahn-Banach theorem. 3. None of the above. Return the answer as a number

category

math

options

  1. 2. Arzelà-Ascoli theorem and Banach fixed-point theorem
  2. 1. Arzelà-Ascoli theorem and Riesz representation theorem
  3. 1.0
  4. 3.0
  5. 2. Sobolev embedding theorem and Riesz representation theorem
  6. 1. Poincare inequality and Banach fixed-point theorem
  7. 2. Poincare inequality and Hahn-Banach theorem
  8. 2.0
  9. 1. Sobolev embedding theorem and Lax-Milgram theorem
  10. All of the above

src

theoremQA-Math

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Source and history

upstream_id / Original problem ID
8418
task_id
38b17b48-8461-5d3f-befb-f97715457bc0
task_key
default--test--8418
task_revision_id
1
config / split
default / test

Official source

initial import