Topic
Poincare Inequality
Courses
Courses whose questions include this topic tag.
Questions
This topic is tagged for 6 questions
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Fall 2024 • Question 1
State (carefully) and prove the standard \(L^2\) Poincaré inequality in a smooth bounded domain \(U\).
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Summer 2024 • Question 2
State (carefully) and prove the standard \(L^2\) Poincaré inequality in a smooth bounded domain \(U\).
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Fall 2013 • Question 9
Formulate and derive the uncertainty principle. Is its inequality optimal? Explain.
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Spring 2013 • Question 9
Formulate and derive the uncertainty principle. Is its inequality optimal?
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Fall 2004 • Question 9
Laplace Equation. Let \(\Omega \subset \mathbb {R}^n\) be a domain. Let \(\Omega _1\) be a bounded subdomain with \(\overline {\Omega }_1\subset \Omega \).
- State the Harnack Inequality for harmonic functions.
- Let \(\{u_n\}\) be a monotone increasing sequence of nonnegative harmonic functions on \(\Omega \). Suppose that for some point \(\xi \in \Omega _1\), the sequence of real numbers \(\{u_n(\xi )\}\) is bounded. Prove that \(\{u_n\}\) converges uniformly on \(\Omega _1\) to a harmonic function. You may use standard theorems about convergence of sequences of harmonic functions without proof.
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Fall 1997 • Question A.4
Let \(\{\phi_i\}\) be a (finite or infinite) orthonormal set in a Hilbert space. Prove that for every vector \(f\) in the space \[ \lVert f\rVert^2\le \sum_i |\langle f,\phi_i\rangle|^2. \] If inequality in above equation is replaced by strict equality, what can be said about the set \(\{\phi_i\}\)?