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Math / Partial Differential Equations: An Introduction 2 / Chapter 7 / Problem 25

Partial Differential Equations: An Introduction | 2nd Edition | ISBN: 9780470054567 | Authors: Walter A. Strauss

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Let the nonconstant function u(x) satisfy the inequality u

Chapter 7, Problem 25

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QUESTION:

Let the nonconstant function satisfy the inequality in a domain in three dimensions. Prove that it cannot assume its maximum inside. This is the maximum principle for subharmonic functions. (Hint: Let , and let denote restricted to the boundary . Let be any ball and let be its centre. Use (11) and (16) together with (7.3.7) in the ball . Show that is at most the average of on .

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QUESTION:

Let the nonconstant function satisfy the inequality in a domain in three dimensions. Prove that it cannot assume its maximum inside. This is the maximum principle for subharmonic functions. (Hint: Let , and let denote restricted to the boundary . Let be any ball and let be its centre. Use (11) and (16) together with (7.3.7) in the ball . Show that is at most the average of on .

ANSWER:

Step 1 of 3

Green's Function: The Green's Function for any half space is given by:

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