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I am reading a paper that seems to provide a solution for the problem I am facing but being unfamiliar with variational calculus I get lost in notation.

I am trying to derive the weak form from the strong form in the following problem.

Solving for $u(t,x)$ for $ (t,x) in [0,T] times R^d $.
The set $A in R^d$ is open with boundary $partial A$.

The strong form is as follows:
$$ frac{partial u}{partial t}(t,x) - frac{1}{2} sum_i sum_j a_{ij}(x) frac{partial^2 u(t,x)}{partial x_i partial x_j} - sum_i b_i(x) frac{partial u(t,x)}{partial x_i} = 0;,; on ; (t,x) in [0,T] times A, $$
$$ u(0,x) = 1, ; x in A, $$
$$ u(t,x) = 0, ; x in partial A,; t > 0$$

The paper indicates that the weak form is as follows:
$$ frac{d u}{d t}(u(t,.),v) + g(u(t,.),v) = 0, forall v in H_0^1(A), $$
$$ u(0,.) = 1, $$
where $ g(u(t,.),v) = frac{1}{2} (a nabla u(t,.), nabla v) - left( (b-text{div } a)nabla u,vright) $.

I assume $ (a,b) = int_A a(x) b(x) dx $.

This seems to be a classical result, the issue is that I am not familiar with the notation, nor with tensor/variational calculus. I assume that it involves a multivariate integration by part, which is foreign to me.

• how to derive $g(u(t,.),v)$ ?


• what is the divergence of the matrix-valued $a$ ?

I get the part with $b$: $int_A left( sum_i b_i(x) frac{partial u(t,x)}{partial x_i} right) v(x) dx = (bnabla u,v)$.

The problem is the part with $a$.

Thanks a lot for any help !

Source of the problem:

P. Patie, C. Winter, (2008) "First exit time probability for multidimensional diffusions: A PDE-based approach"

I hope this is correct, there is still some uncertainty for some parts.

Focusing on:
$$ sum_i sum_j int a_{ij}(x) partial_{ij} u(x) v(x) dx tag{1} label{1}$$

For a given dimension (e.g. $x_j$) we do an IBP.

We have $[a_{ij}(x) partial_{i} u(x) v(x)]_{partial A}=0$, since $v(x) = 0$ at the boundary.
$$ int a_{ij}(x) partial_{ij} u(x) v(x) dx = [a_{ij}(x) partial_{i} u(x) v(x)]_{partial A} - int ( partial_j a_{ij}(x)v(x)+ a_{ij}(x)partial_j v(x))partial_i u(x)dx \
= - int partial_j a_{ij}(x)v(x) partial_i u(x) dx - int a_{ij}(x)partial_j v(x)partial_i u(x)dx
$$

Now we sum the two terms over $i$ and $j$.

For the first term we have:
$$
begin{align}
sum_i sum_j int partial_j a_{ij}(x)v(x) partial_i u(x) dx &
= int left( sum_i left( sum_j partial_j a_{ij}(x)right) partial_i u(x)right) v(x) dx \
& = int left( sum_i (text{div } a)_i partial_i u(x)right) v(x) dx \
& = (text{div } a nabla u,v)
end{align}
$$
where $ (text{div } a)_i = left( sum_j partial_j a_{ij}(x) right)_i $.

For the second term we have:
$$
begin{align}
sum_i sum_j int a_{ij}(x)partial_j v(x)partial_i u(x)dx
& = int sum_j left( sum_i a_{ij}(x) partial_i u(x) ) right) partial_j v(x) dx \
& = ( anabla u , nabla v)
end{align}
$$

Plugging this in $(ref{1})$, we get:
$$ (1) = - (text{div } a nabla u,v) - ( anabla u , nabla v). tag{2} $$

Coming back to the original term, we wanted:
$$ - int left( frac{1}{2} sum_i sum_j a_{ij}(x) partial_{ij} u(x) + sum_i b_i(x) partial_i u(x) right) v(x) dx. tag{3} $$
Using the fact that $ int sum_i b_i(x) partial_i u(x) v(x) dx = (bnabla u,v),; $ and (2), we get:
$$ begin{align}
(3) & = frac{1}{2} (text{div } a nabla u,v) + frac{1}{2} ( anabla u , nabla v) - (bnabla u,v) \
& = frac{1}{2} ( anabla u , nabla v) - ( (b - frac{1}{2} text{div } a ) nabla u, v )
end{align}
$$

This is different from what the paper gives, I have an additional "$frac{1}{2}$" (who made an error ?):
$$ begin{align}
(g) & = frac{1}{2} ( anabla u , nabla v) - ( (b - text{div } a ) nabla u, v )
end{align}
$$

So I see that it mostly works if we have a nice bounded set such as $A={l_i<x_i<u_i;;i=1,...,d}$.

But I am not sure what happens if it is unbounded on one side: $A={x_i<u_i;;i=1,...,d}$.

Or worse, it is a weird boundary: $A={x_i-x_j<u_i;;i,j=1,...,d}$.
Feel free to comment, I will edit the answer accordingly.