Green's identities for Laplacian-squared

1. Feb 13, 2013

TaPaKaH

1. The problem statement, all variables and given/known data
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 $u,v\in C^\infty(U)$.

3. 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.

2. Feb 14, 2013

TaPaKaH

Using the following identity$$\int_U\frac{\partial^2 u}{\partial x_i^2}vdx=-\int_{\partial U}\frac{\partial u}{\partial x_i}vn_i dS+\int_{\partial U}u\frac{\partial v}{\partial x_i}n_idS-\int_Uu\frac{\partial^2v}{\partial x_i^2}dx$$
(where $n_i$ is i-th coordinate of the orthogonal vector) I did some writing out
$$\int_U(\bigtriangleup^2u)vdx=\int_U \bigtriangleup\left(\sum_{j=1}^n\frac{\partial^2u}{\partial x_j^2}\right)vdx=\int_U\left(\sum_{i=1}^n\sum_{j=1}^n\frac{\partial^4u}{\partial x_i^2\partial x_j^2}\right)vdx=\sum_{i=1}^n\sum_{j=1}^n\int_U \frac{\partial^4u}{\partial x_i^2\partial x_j^2}vdx=$$
$$=\sum_{i=1}^n\sum_{j=1}^n\left(-\int_{\partial U}\frac{\partial^3u}{\partial x_i\partial x_j^2}vn_idS+\int_{\partial U}\frac{\partial^2u}{\partial x_j^2}\frac{\partial v}{\partial x_i}n_idS-\int_U\frac{\partial^2u}{\partial x_j^2}\frac{\partial^2v}{\partial x_i^2}dx\right)=$$
$$=\sum_{i=1}^n\sum_{j=1}^n\left(\int_{\partial U}\left(\frac{\partial^2u}{\partial x_j^2}\frac{\partial v}{\partial x_i}-\frac{\partial^3u}{\partial x_i\partial x_j^2}v\right)n_idS+\int_{\partial U}\frac{\partial u}{\partial x_j}\frac{\partial^2v}{\partial x_i^2}n_jdS-\int_{\partial U}u\frac{\partial^3v}{\partial x_i^2\partial x_j}n_jdS+\int_Uu\frac{\partial^4v}{\partial x_i^2\partial x_j^2}dx\right)=$$
$$=\sum_{i=1}^n\sum_{j=1}^n\int_{\partial U}\left(\left(\frac{\partial^2u}{\partial x_j^2}\frac{\partial v}{\partial x_i}-\frac{\partial^3u}{\partial x_i\partial x_j^2}v\right)n_i+\left(\frac{\partial u}{\partial x_j}\frac{\partial^2v}{\partial x_i^2}-u\frac{\partial^3v}{\partial x_i^2\partial x_j}\right)n_j\right)dS+\sum_{i=1}^n\sum_{j=1}^n \int_Uu\frac{\partial^4v}{\partial x_i^2\partial x_j^2}dx=$$
$$=\sum_{i=1}^n\sum_{j=1}^n\int_{\partial U}\left(\left(\frac{\partial^2u}{\partial x_j^2}\frac{\partial v}{\partial x_i}-\frac{\partial^3u}{\partial x_i\partial x_j^2}v\right)n_i+\left(\frac{\partial u}{\partial x_j}\frac{\partial^2v}{\partial x_i^2}-u\frac{\partial^3v}{\partial x_i^2\partial x_j}\right)n_j\right)dS+\int_Uu(\bigtriangleup^2v)dx$$

Could anyone suggest a way to simplify the double sum in the last expression?

EDIT: nevermind, seem to have worked it out myself

Last edited: Feb 14, 2013