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[SOLVED] Laplace-Fourier Analysis

dwsmith

Well-known member
Feb 1, 2012
1,673
Prove that the steady state temperature in a disk is never higher than the highest temperature on the boundary.
Prove that for a steady state temperature distribution in a disk, the coldest spot is on the boundary, unless the temperature is constant throughout the disk.



How is this part suppose to help? I don't see it.
$$
\int_{-\pi}^{\pi}P(r,\theta)d\theta = 1
$$
Recall from calculus that
$$
m(b - a)\leq\int_a^bf(x)dx\leq M(b - a)
$$
where $M$ and $m$ are the maximum and minimum, respectively on the interval $[a,b]$.