Derive the three-dimensional maximum principle from the

Chapter 7, Problem 1

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QUESTION:

Derive the three-dimensional maximum principle from the mean value property.

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QUESTION:

Derive the three-dimensional maximum principle from the mean value property.

ANSWER:

Step 1 of 2

The maximum principle says that, any solid region considered as a ,  is a non-constant harmonic function and it cannot take maximum value inside it but on bdy.

Let’s consider that  is a non-constant harmonic function in it takes maximum value

inside .

According to the mean value property, we have

                                                   

For all , then , for all  since u is continuous and M is its Maximum.

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