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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2015 • Question 5

Question

Math 6420 • PDE Fall 2015 • Question 5

Consider the distribution \(u\in \mathcal {D}'(\mathbb {R})\) defined for \(\phi \in \mathcal {D}(\mathbb {R})\) by \[ \langle u,\phi \rangle =\int _{-\infty }^{\infty }\lvert x\rvert \phi (x)\,dx. \] Find its derivative \(\partial u\).

Raw LaTeX

Consider the distribution \(u\in\mathcal{D}'(\mathbb{R})\) defined for
\(\phi\in\mathcal{D}(\mathbb{R})\) by
\[
  \langle u,\phi\rangle=\int_{-\infty}^{\infty}\lvert x\rvert\phi(x)\,dx.
\]
Find its derivative \(\partial u\).

Topic Tags

Distributions and Weak Solutions Linear Programming and Optimization Real Integral Evaluation by Complex Methods