Question

A (nonconstant) harmonic function takes its maximum value and its minimum value

Answer 1

Answer:

Explanation:

Consider that F (any function) <0 .

u(x,y) is a coontinuous function in the closed interval or region R.

Let us consider a point (p,q) that is inside the region and it is a maximum point.

Then it should be must

uxx (p,q) <0 where uxx means double differentiation

and uy(p,q) >0

Since ux(p,q) = 0 = uy(p,q) where ux and uy means single differentiation with respect to x and y respectively.

Say, Maximum limits of the region is T

therefore q<T

then uy (p,q) = 0 if q<T

if q = T then

point (p,q) = (p,T) will be on the boundary of R then we claim that

uy(p,q) >0

Similarly for the minimum also it will work