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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2008 • Question 5

Question

Math 6510 • Differentiable Manifolds Fall 2008 • Question 5

Let \(M\) be a compact oriented Riemannian manifold with boundary. Suppose that \(u\) and \(v\) are harmonic functions that are equal on \(\partial M\). Show that \(u=v\) on \(M\). Recall that \(u\) is harmonic on \(M\) if \(\Delta u=\operatorname {div}(\nabla u)=0\) on \(M\).

Raw LaTeX

Let \(M\) be a compact oriented Riemannian manifold with boundary. Suppose
that \(u\) and \(v\) are harmonic functions that are equal on \(\partial M\). Show that
\(u=v\) on \(M\). Recall that \(u\) is harmonic on \(M\) if
\(\Delta u=\divop(\nabla u)=0\) on \(M\).

Topic Tags

Homotopy and Degree on Manifolds Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Stokes' Theorem and Integration on Manifolds