= 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 $v$ satisfies
$$
-\Delta v\leq f\quad\text{inside},\qquad v\leq g\quad\text{on the boundary}.
$$
If a suitable <comparison principle for harmonic functions> holds and $u$ is a solution, then $v\leq u$. 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 $-\Delta v=-2n\leq0$, 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.
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