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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2019 • Question 5

Question

Math 6420 • PDE Fall 2019 • Question 5

Let \(L\) be a partial differential operator of the form: \[ L=\sum _{j=0}^{m-1} c_j\partial ^j, \] where the \(c_j\) are arbitrary constants. Show that \(x^m(L\delta )=0\) as distributions in \(\mathcal {D}'(\mathbb {R})\). Here \(\delta \in \mathcal {D}'(\mathbb {R})\) is the usual “delta function”.

Raw LaTeX

Let \(L\) be a partial differential operator of the form:
\[
  L=\sum_{j=0}^{m-1} c_j\partial^j,
\]
where the \(c_j\) are arbitrary constants. Show that \(x^m(L\delta)=0\) as distributions in
\(\mathcal{D}'(\mathbb{R})\). Here \(\delta\in\mathcal{D}'(\mathbb{R})\) is the usual
``delta function''.

Topic Tags

Distributions and Weak Solutions Fundamental Solutions Linear Programming and Optimization Maximum Modulus Principle