I am reading Qing Han,& Fanghua Lin’s elliptic partial differential equations. The book had the following variable transformations. (proof of theorem 4.1)
Suppose that $u\in H^1_0(B_1)$ is a subsolution in the following sense: $$\int_{B_1}a_{ij}D_iuD_j\varphi+cu\varphi\leq \int_{B_1}f\varphi\quad \text{for any }\varphi\in H^1_0(B_1)\text{ and }\varphi\geq 0\text{ in }B_1. $$ Take any $R\leq 1$. Define $$\tilde{u}(y)=u(Ry)\quad\text{ for }y\in B_1.$$ It is easy to see that $\tilde{u}$ satisfies the following equation: $$\int_{B_1}\tilde{a}_{ij}D_i\tilde{u}D_j\varphi+\tilde{c}\tilde{u}\varphi\leq \int_{B_1}\tilde{f}\varphi\quad \text{for any }\varphi\in H^1_0(B_1)\text{ and }\varphi\geq 0\text{ in }B_1. $$ where $$\tilde{a}(y)=a(Ry),\quad \tilde{c}(y)=R^2c(Ry),\quad \tilde{f}(y)=R^2f(Ry),$$ for any $y\in B_1.$
I thought it was a simple variable transformation, but there were a few things I did not understand. Why is the integral range still $B_1$ instead of $B_{R}$ ? Also, why do we need $R^2$ ? I would like someone to explain this to me.