When proving weak maximum principle, for example $\Delta u\geq 0$, to get a contradiction, we need to introduce an auxiliary function e.x. $v=\epsilon e^{x_1}$, to make $\Delta (u+v)>0$. I am wondering what is the geometric sense behind introducing this auxiliary function?
2026-03-25 03:07:16.1774408036
The geometric intepretation behind introducing auxiliary function in weak maximum principle
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What is the geometric sense of $\Delta u >0$ ?