Answered: How to show that v(x) = ln|x| is a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

How to show that v(x) = ln|x| is a subharmonic function in R^n\{0} ?

Expert Solution

Step 1: Explanation

To show that $v not stretchy left parenthesis x not stretchy right parenthesis equals ln invisible function application not stretchy vertical line x not stretchy vertical line$ is a subharmonic function in $real numbers to the power of n set minus not stretchy left curly bracket 0 not stretchy right curly bracket$ , we need to prove that it satisfies the subharmonic inequality. A function $u$ is subharmonic if, for any ball $B subscript r not stretchy left parenthesis x subscript 0 not stretchy right parenthesis$ (centered at $x subscript 0$ ) contained in $real numbers to the power of n set minus not stretchy left curly bracket 0 not stretchy right curly bracket$ , the following inequality holds:

$u not stretchy left parenthesis x subscript 0 not stretchy right parenthesis less or equal than fraction numerator 1 over denominator not stretchy vertical line B subscript r not stretchy vertical line end fraction integral subscript B subscript r not stretchy left parenthesis x subscript 0 not stretchy right parenthesis end subscript u not stretchy left parenthesis x not stretchy right parenthesis blank d x comma$

where $not stretchy vertical line B subscript r not stretchy vertical line$ denotes the volume (measure) of the ball.