Subsolution

A subsolution satisfies a one-sided version of a partial differential equation, with the inequality convention tied to the operator's sign. For the Dirichlet problem $-\Delta u=f$ with boundary data $g$, a classical subsolution $v$ satisfiesIf a suitable comparison principle for harmonic functions holds and $u$ is a solution, then $v\le u$. Supersolutions reverse both inequalities.

In the unit ball of ${\mathbb{R}}^n$ with $f=g=0$, take $v(x)=|x{|}^2-1$. It vanishes on the boundary and $-\Delta v=-2n\le 0$, so it lies below the zero harmonic solution. The maximum principle explains the general comparison: $v-u$ is subharmonic and nonpositive on the boundary.

Weak subsolutions interpret the inequality against nonnegative test functions; viscosity subsolutions instead use touching smooth test functions. These are specified solution notions, not an automatic pointwise interpretation for a nonsmooth function.