- #1
kajzlik
- 3
- 0
Hello guys, suppose we have an eigenvalue problem
[tex]
\left\{
\begin{array}{ll}
u'' + λu = 0, \quad x \in (0,\pi) \\
u(0)=0 \quad \\
\left( \int_0^\pi \! {(u^+)}^2 \, \mathrm{d}x \right)^{\frac{1}{2}} = \left( \int_0^\pi \! {(u^-)}^4 \, \mathrm{d}x \right)^\frac {1}{4}
\end{array}
\right.
[/tex]
where [tex] u^+, u^-[/tex] is positive, negative part of function respectively.
I'm having troubles with case when λ> 0. Is there any way how to simplify or express these parts of function ? I've tried some (analytic) brute force methods, tried to simplify that first but still no valuable result.
Thanks
[tex]
\left\{
\begin{array}{ll}
u'' + λu = 0, \quad x \in (0,\pi) \\
u(0)=0 \quad \\
\left( \int_0^\pi \! {(u^+)}^2 \, \mathrm{d}x \right)^{\frac{1}{2}} = \left( \int_0^\pi \! {(u^-)}^4 \, \mathrm{d}x \right)^\frac {1}{4}
\end{array}
\right.
[/tex]
where [tex] u^+, u^-[/tex] is positive, negative part of function respectively.
I'm having troubles with case when λ> 0. Is there any way how to simplify or express these parts of function ? I've tried some (analytic) brute force methods, tried to simplify that first but still no valuable result.
Thanks