Estimate and evaluate
Assume that : $int_0^1 |u|^2 = 5 $.
Provide an estimate for $ int_0^h |u| $ being $ 0
Provide an estimate for $ int_0^h |u| $ being $ 0
Risposte
"Eredir":
Is this solution correct?
Yes, it is
Is this solution correct? 
Using Cauchy-Schwarz inequality we get $\int_0^h|u(x)|dx <= (\int_0^h|u(x)|^2dx)^(1/2)(\int_0^h1^2dx)^(1/2) = sqrt(h)(\int_0^h|u(x)|^2dx)^(1/2)$.
Now since $|u(x)|^2$ is non-negative we have $sqrt(h)(\int_0^h|u(x)|^2dx)^(1/2) <= sqrt(h)(\int_0^1|u(x)|^2dx)^(1/2)=sqrt(5h)$.
Our estimate is then $\int_0^h|u(x)|dx <= sqrt(5h)$, and to get $\int_0^h|u(x)|dx < 1/2$ we simply need to put $h<1/20$.
Now since $|u(x)|^2$ is non-negative we have $sqrt(h)(\int_0^h|u(x)|^2dx)^(1/2) <= sqrt(h)(\int_0^1|u(x)|^2dx)^(1/2)=sqrt(5h)$.
Our estimate is then $\int_0^h|u(x)|dx <= sqrt(5h)$, and to get $\int_0^h|u(x)|dx < 1/2$ we simply need to put $h<1/20$.
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