Integrating until symmetric bilinear form

[TheFerruccio]

Homework Statement

I am looking for some quick methods to integrate while leaving each step in its vector form without drilling down into component-wise integration, and I am wondering whether it is possible here.

Suppose I have a square domain over which I am integrating two functions w and v. Using integration by parts, what steps can I use to ensure that I end up with a symmetric blinear form + boundary terms?

Homework Equations

$\exists$ a region $\Omega$ enclosed by $\Gamma$ and on $\Omega$ $$\int_\Omega{Tv\nabla^2 w d\Omega}=\int_\Omega{v p(x,y) d\Omega}$$

The Attempt at a Solution

Using integration by parts, whereby I feel as if I am completely guessing here:
$$T\left[v\nabla w\right]_\Gamma-T\int_\Omega{\nabla v \cdot \nabla w dV} = \int_\Omega{v p(x,y) d\Omega}$$

Is this the correct symmetric bilinear form that I am looking for? How would that simplified boundary term expression be evaluated explicitly?

 

[TheFerruccio]

Unfortunately not. I have come to the conclusion that there is no one here who knows this particular material. It is a blend of math and engineering and computation. I will update the threads with solutions once I find them. I eventually do, but I have never gotten a single question of this area of material answered on these forums, which is a first.