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Let $u_1$ and $u_2$ be two Dirichlet functions; hence both attain their maximum and minimum values on the boundary of the domain $D$ (let us call the boundary $B$).

My book says the following:

Let $v=u_1-u_2$. Then $V$ also attains its minimum and maximum values on $B$ as a consequence of the minimum-maximum principle.

I don't see why this has to be correct. I feel $v$ may not attain its minimum and maximum values on $B$- it may attain them in the interior of $D$ itself.

algebraically_speaking
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  • I believe when you say dirichlet function, you say harmonic functions defined in $\overline{\Omega}$ ($\Omega$ a bounded smooth domain of $R^n$). The diference of harmonic functions is harmonic. Then by the maximum/minimum principle the diference function satisfies what your book says. – math student Apr 21 '14 at 02:14
  • the only exception for the maximum/minimum principle is when the function is constant. – math student Apr 21 '14 at 02:15
  • @algebraically_speaking: Unless you define what is to be meant by "Dirichlet function", it is likely to be taken for its celebrated namesake http://mathworld.wolfram.com/DirichletFunction.html – mkl314 Apr 21 '14 at 11:42

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