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TaPaKaH
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Homework Statement
a) Derive Green's identities in local and integral form for the partial differential operator ##\triangle^2##.
b) Compute the adjoint operator ##(\triangle^2)^*##.
2. Relevant information
##U\subset\mathbb{R}^n##, ##u:U\to\mathbb{R}##, differential operator ##Lu##.
In the lecture notes it is given that:
First Green's identity: ##\int_U\left[(Lu)v-u(L^*v)\right]dx=0## for any U and ##u,v\in C_0^\infty(U)##.
Second Green's identity: ##\int_U\left[(Lu)v-u(L^*v)\right]dx=\int_{\partial U}QndS## for some field Q and ##u,v\in C^\infty(U)##.
The Attempt at a Solution
I haven't done PDEs in a long while, do I get it right that the order of actions to solve the exercise would be:
1) Assume ##u,v\in C_0^\infty(U)##, by partial integration obtain ##\int_U(\triangle^2u)vdx= \int_Uu(\triangle^2v)dx## (am not fully sure if this holds, will need to verify) meaning via 1st Green's identity that ##(\triangle^2)^*=\triangle^2##.
2) Plug this into 2nd Green's identity, assume ##u,v\in C^\infty(U)## and get the expression for Q.