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4.* Let u be a harmonic function in R" and suppose that lu(x)/*dx < oo. R" Show that u = 0. (Hint: Write the mean value property in a ball of radius r and use the Cauchy- Schwarz inequality given in Question 3.)

4.* Let u be a harmonic function in R" and suppose that lu(x)/*dx < oo. R" Show that u = 0. (Hint: Write the mean value property in a ball of radius r and use the Cauchy- Schwarz inequality given in Question 3.)

Advanced Engineering Mathematics
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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4.* Let u be a harmonic function in R" and suppose that lu(x)/*dx < oo. R" Show that u = 0. (Hint: Write the mean value property in a ball of radius r and use the Cauchy- Schwarz inequality given in Question 3.)
Transcribed Image Text:4.* Let u be a harmonic function in R" and suppose that lu(x)/*dx < oo. R" Show that u = 0. (Hint: Write the mean value property in a ball of radius r and use the Cauchy- Schwarz inequality given in Question 3.)
3. Let U C R" and u, v e L²(U). Prove the Cauchy-Schwarz inequality, u(х)u(х) dx| < |u(x)/² dx (Hint: Let A := (u, v)/({v, v)) and note that ||u – Av||² > 0.)
Transcribed Image Text:3. Let U C R" and u, v e L²(U). Prove the Cauchy-Schwarz inequality, u(х)u(х) dx| < |u(x)/² dx (Hint: Let A := (u, v)/({v, v)) and note that ||u – Av||² > 0.)
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