I know we can prove the minimum principle by using the maximum principle, by replacing $u(x,y)$ by $-u(x,y)$, but I keep getting stuck. Can someone help me figure out how to continue?
So we have Laplace's equation $$u_{xx}+u_{yy} =0$$ and to prove the minimum principle, we let $$v(x,y) = -u(x,y) + \epsilon (x^2+y^2)$$ and let $\min u(x,y) = m$ and $\min (x^2+y^2) =l$.
From this, we get that $$v(x,y) \geq -m + \epsilon l$$. This is where I keep getting stuck. I'm not sure if I'm doing this correctly or incorrectly, but I can't get further than this to prove that $u(x,y) \geq m$ as $\epsilon$ goes to $0$.
Can someone help me figure out where I'm going wrong or what to do ahead? (Also really sorry for this basic question, but please bear with me, I'm really trying ;-;)