I found the following question in a book only with one sentence. "This question can be solved by an elementary way. Note that the following two are false: (1) If a function is bounded from below, then it has minimum value. (2) A monotone decreasing sequence reaches a negative value."

Question: Let $m,n$ be integers. Supposing that a function $f(m,n)$ defined by $m,n$ satisfies the following two conditions, then prove that $f(m,n)$ is a constant.

1. $f(m,n)\ge0$.

2. $4f(m,n)=f(m-1,n)+f(m+1,n)+f(m,n-1)+f(m,n+1)$.

I suspect this question can be solved by a geometric aspect. I've tried to prove this, but I'm facing difficulty. Could you show me how to prove this?

  • $\begingroup$ It is a discrete version of Laplace equation $\nabla^{2}\Phi = 0$ which leads to $\int\Psi\nabla^{2}\Psi\,{\rm d}V = 0$ and to $\int\left\vert\nabla\Psi\right\vert^{2} = 0$ which involves an integration by parts. The discrete version requires "Abel Summations by part" identity ( en.wikipedia.org/wiki/Summation_by_parts ). $\endgroup$ – Felix Marin Sep 6 '13 at 7:29
  • $\begingroup$ Where did you get this problem? It feels familiar to me for some reason. $\endgroup$ – davidlowryduda Sep 6 '13 at 7:47
  • $\begingroup$ @mixedmath: I found this in the book which a friend of mine has, but I don't know its name. I'll ask him. $\endgroup$ – mathlove Sep 6 '13 at 7:53
  • $\begingroup$ @FelixMarin: Thank you for nice information. $\endgroup$ – mathlove Sep 6 '13 at 7:57
  • $\begingroup$ @mathlove You're welcome. $\endgroup$ – Felix Marin Sep 6 '13 at 23:44

A probabilistic proof is given at the bottom of this page.


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