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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1997 • Question 10

Question

Math 6410 • ODE Fall 1997 • Question 10

Show that there is a constant \(c<\infty \) depending only on \(n\) and \(r\) so that \[ \sup \{\lvert Du(x)\rvert :x\in \overline {B_r(0)}\}\le c\sup \{\lvert u(x)\rvert :x\in \partial B_{2r}(0)\}. \] for all harmonic functions \(u\) on \(B_{2r}(0)\).

Raw LaTeX

Show that there is a constant \(c<\infty\) depending only on \(n\) and \(r\) so
that
\[
  \sup\{\lvert Du(x)\rvert:x\in\overline{B_r(0)}\}\le
  c\sup\{\lvert u(x)\rvert:x\in\partial B_{2r}(0)\}.
\]
for all harmonic functions \(u\) on \(B_{2r}(0)\).

Topic Tags

Laplace Equation Variation of Constants Formula