Comparison principle

The comparison principle refers to the general concept that a subsolution to an elliptic equation stays below a supersolution of the same equation. It known to hold under a great generality of assumptions.

The comparison principle can also be understood as the fact that the difference between a subsolution and a supersolution satisfies the maximum principle. The uniqueness of the solution of the equation is an immediate consequence.

General statement

The two statements below correspond to the comparison principle for elliptic and parabolic equations with Dirichlet boundary conditions. The main difference with the local case, is that for nonlocal equations the Dirichlet condition has to be taken in the whole complement of the domain $\Omega$ instead of only the boundary.

Other boundary conditions require appropriate modifications.

Elliptic case

We say that an elliptic equation $Iu=0$ satiesfies the comparison principle if the following statement is true.

Given two functions $u : \R^n \to \R$ and $v : \R^n \to \R$ such that $u$ and $v$ are upper and lower semicontinuous in $\overline \Omega$ respectively, where $\Omega$ is an open domain, $Iu \geq 0$ and $Iv \leq 0$ in the viscosity sense in $\Omega$, and $u \leq v$ in $\R^n \setminus \Omega$, then $u \leq v$ in $\Omega$ as well.

Parabolic case

We say that a parabolic equation $u_t - Iu=0$ satiesfies the comparison principle if the following statement is true.

Given two functions $u : [0,T] \times \R^n \to \R$ and $v : [0,T] \times\R^n \to \R$ such that $u$ and $v$ are upper and lower semicontinuous in $[0,T] \times \overline \Omega$ respectively, $Iu \leq 0$ and $Iv \geq 0$ in the viscosity sense in $(0,T] \times \Omega$, and $u \leq v$ in $(\{0\} \times \R^n) \cup ([0,T] \times (\R^n \setminus \Omega))$, then $u \leq v$ in $[0,T] \times \Omega$ as well.