D let f be a positive continuous 2 periodic function with...

Name: d let f be a positive continuous 2 periodic function with harmonic extension ur a prove that ur 0 1r b prove harnacks inequality u0 ur u0 1 r 1r 48258
Uploaded: 2021-12-21T09:57:38-08:00
Duration: 4 min 10 s
Channel: Harmender Singh Yadav
Description: d let f be a positive continuous 2 periodic function with harmonic extension ur a prove that ur 0 1r b prove harnacks inequality u0 ur u0 1 r 1r 48258

D. Let f be a positive continuous 2π-periodic function with harmonic extension u(r, θ). (a) Prove that u(r, θ) > 0. 1+r (b) Prove Harnack's inequality: u(0,θ) ≤ u(r, θ) ≤ u(0,θ). 1 r 1+r

Transcript

00:01 In this question, it is given that the function f is continuous on the close interval a to b, and f of x is always greater than or equal to 0 on the interval, that is x belonging to the interval a to b.

00:15 Now, we need to prove that the integral of the function f of x for the limit a to b is also greater than or equal to 0.

00:22 So, from the definition of the definite integral, we know that that is the integral of function f of x dot d x for the limit a to b is equal to actually that is the limit we can write it as the that is sum of limit limit of the sum that is limit is n tends to infinite for that is summation of f of u i dot delta x that is d x or delta x here and where i is from 1 to n so this is the from the definition of definite integrals we can write the integral of f of x definite integral of x for the limit a to b as the limit of sum sum of f of ui times delta x where i is from 1 to n and n tends to infinite this is the limit now we know as the function f of x is greater than 0 and also that is delta x is also greater than 0 so we will have as f of x greater than r equal to 0 and delta x will also be greater than 0 so we will have that the summation of this f of ui delta x will also greater than 0 so from here we get that that is summation for f of ui times delta x for i 1 to n will also be greater than or equal to 0 therefore by the that is properties of the limits we will have this that is the summation of that is limit of limit for n tends to infinite that is summation of f of ui dot delta x for i equals to to 1 to n this will be greater than or equal to limit n tends to infinite on 0, the limit on 0 which is also equal to 0 here.

02:53 So on the left hand side we know that this definition actually corresponds to the integral of that is f of x .d .x we can apply the limits as a to b...

Question

D. Let f be a positive continuous 2π-periodic function with harmonic extension u(r, θ). (a) Prove that u(r, θ) > 0. 1+r (b) Prove Harnack's inequality: u(0,θ) ≤ u(r, θ) ≤ u(0,θ). 1 r 1+r

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