I am considering taking the derivative of the function $$F(\mathbb{x_1},\mathbb{x_2},\mathbb{x_3}) = \displaystyle \int_{V_1} ||x-\mathbb{x_1}||\phi(x)\,dx + \int_{V_2} ||x-\mathbb{x_2}||\phi(x)\,dx + \int_{V_3} ||x-\mathbb{x_3}||\phi(x)\,dx$$
where $\mathbb{x}_k \in \mathbb{R^n}$, $\phi(x)$ has a compact support, and the regions $V_i$ are defined as: $$V_i = \{x: ||x-\mathbb{x_i}|| \le ||x-\mathbb{x}_j||, j \neq i \}.$$
Since $\mathbb{x}_k \in \mathbb{R^n}$, I plan to take the partial derivative with respect to each component of $\mathbb{x}_k,$ say $\mathbb{x}_k^i$ where $i \in \{1,\dots,n\}$. In other words, $\mathbb{x}_k = (\mathbb{x}_k^1,\dots,\mathbb{x}_k^n)$.
Let's suppose that I am going to take the derivative with respect to $\mathbb{x_1}^i$. My problem is that not only is the integrand $||x-\mathbb{x_1}||$ a function of $\mathbb{x_1}^i$ but so are $V_1,V_2,V_3$.
I learned this trick in my PDE class wherein for $V_1$, the region $ ||x-\mathbb{x_1}|| \le ||x-\mathbb{x}_j|| $ can be shifted so that it is centered at $0$ instead of $\mathbb{x_1}$ thereby transfering the dependence on $\mathbb{x_1}^i$ from the domain of integration to the integrand.
But, my problem is with the regions $V_2$ and $V_3$, i.e. the latter 2 integrals above. For $V_2$ for example, the region is defined as $\{x: ||x-\mathbb{x_2}|| \le \text{min} (||x-\mathbb{x}_1||,||x-\mathbb{x}_3||) \}$. This is where I am currently stuck. How do I transfer the dependence on $\mathbb{x}_1^i$ from the domain of integration to the integrand?
This problem has been tackled in the paper http://link.springer.com/chapter/10.1007%2FBFb0008901#page-1 but the details are not provided.
Any suggestions on how to proceed?
For your case, the contribution from changes of $V_i$ are zero.
First, the integral can be rewritten as
$$\int f(x)\phi(x) dx \quad\text{ where }\quad f(x) = \min\left(\|x-x_1\|, \| x - x_2\|, \| x - x_3\|\right)$$ If you move $x_i$ for a distance $\epsilon$, the change of $\| x - x_i\|$ and hence $f(x)$ will be bounded by $\epsilon$.
Under such a change of $x_i$, the boundary between $V_i$ and $V_j, j \ne i$ moves for an amount of order $$\frac{\epsilon}{\left|\nabla \left(\|x - x_i\| - \| x - x_j\|\right)\right|}$$
If is easy to see $$\| x - x_i \| = \| x - x_j \| \quad\implies\quad \nabla \left(\|x - x_i\| - \| x - x_j \|\right) \ne 0$$
This means the contribution of the boundary between $V_i$ and $V_j$ to the change of integral will be $O(\epsilon^2 \ell)$ where $\ell$ is the "length" of the boundary inside support of $\phi$. As a consequence, they contribute nothing to the final result of "differentiation under integral sign".