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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Winter 2005 • Question 9

Question

Math 6410 • ODE Winter 2005 • Question 9

Laplace Equation. Let \(\Omega \subset \mathbb {R}^n\) be open. Suppose \(u\in C(\Omega )\) satisfies the mean value property: for every ball \(B_r(\xi )=\{x\in \mathbb {R}^n:\lvert x-\xi \rvert <r\}\) with \(B_r(\xi )\subset \Omega \), \[ u(\xi )=\frac {1}{\omega _n}\int _{\lvert x\rvert =1}u(\xi +rx)\,dS_x, \] where \(\omega _n\) is the measure of the \((n-1)\)-dimensional unit sphere in \(\mathbb {R}^n\). Prove that \(\Delta u=0\) in \(\Omega \).

Raw LaTeX

\textbf{Laplace Equation.}
Let \(\Omega\subset\R^n\) be open. Suppose \(u\in C(\Omega)\) satisfies the mean value
property: for every ball \(B_r(\xi)=\{x\in\R^n:\lvert x-\xi\rvert<r\}\) with
\(B_r(\xi)\subset\Omega\),
\[
  u(\xi)=\frac{1}{\omega_n}\int_{\lvert x\rvert=1}u(\xi+rx)\,dS_x,
\]
where \(\omega_n\) is the measure of the \((n-1)\)-dimensional unit sphere in \(\R^n\). Prove
that \(\Delta u=0\) in \(\Omega\).

Topic Tags

Distributions and Weak Solutions Fundamental Solutions Gronwall Inequality and Continuous Dependence Laplace Equation