Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2006 • Question 6

Question

Math 6410 • ODE Fall 2006 • Question 6

Let \(U\) be an open subset of \(\mathbb {R}^2\). Prove that if \(u\in C^2(U)\) satisfies \[ \Delta u-u=0 \quad \text {in } U, \] then \[ u(x)=\frac {1}{I(r)}\int _{\partial B(x,r)} u\,dS_y \] for every ball \(B(x,r)\subset U\), where \[ I(r)=\frac {1}{\pi }\int _0^\pi e^{r\cos t}\,dt. \] Assume without proof that \(I(r)\) is the unique solution to the modified Bessel equation \[ r\frac {d^2\varphi }{dr^2}+\frac {d\varphi }{dr}-r\varphi =0,\quad r>0, \] which satisfies \(\varphi (0)=1\).

Raw LaTeX

Let \(U\) be an open subset of \(\R^2\). Prove that if \(u\in C^2(U)\) satisfies
\[
  \Delta u-u=0 \quad \text{in } U,
\]
then
\[
  u(x)=\frac{1}{I(r)}\int_{\partial B(x,r)} u\,dS_y
\]
for every ball \(B(x,r)\subset U\), where
\[
  I(r)=\frac{1}{\pi}\int_0^\pi e^{r\cos t}\,dt.
\]
Assume without proof that \(I(r)\) is the unique solution to the modified Bessel equation
\[
  r\frac{d^2\varphi}{dr^2}+\frac{d\varphi}{dr}-r\varphi=0,\quad r>0,
\]
which satisfies \(\varphi(0)=1\).

Topic Tags

Distributions and Weak Solutions Existence and Uniqueness Theorems Fundamental Solutions Poincare Maps and Periodic Solutions