Supersolutions

A supersolution is a function used in the analysis of differential equations to bound the true solution from above. Let Ω be a domain and L a differential operator (such as the Laplacian Δ or a nonlinear operator). For a boundary-value problem Lu = f in Ω with data on the boundary ∂Ω, a function u is called a supersolution if Lu ≥ f in Ω and u satisfies appropriate boundary inequalities (for example u ≥ boundary data on ∂Ω in the Dirichlet setting). In this sense, a supersolution satisfies a differential inequality that is stronger than the original equation and lies above the prescribed boundary values.

In the simplest elliptic case, when L is the Laplacian and the equation is Lu = 0, a supersolution satisfies Δu ≥ 0 in Ω, while a subsolution satisfies Δv ≤ 0. These notions generalize to many operators, including nonlinear ones like the p-Laplacian, where a supersolution meets Δp u ≥ 0 in Ω.

Supersolutions are closely tied to subsolutions and to comparison principles. If v is a subsolution and w is a supersolution with v ≤ w on the boundary ∂Ω, then under suitable regularity, v ≤ w throughout Ω. This principle allows the use of barriers and bounding functions to control the behavior of the actual solution.

Constructive methods often rely on families of subsolutions or supersolutions. Perron’s method, for example, builds a solution as the supremum of subsolutions or the infimum of supersolutions that respect the boundary data. Barrier functions provide explicit supersolutions to constrain the solution near the boundary.

The concept extends to time-dependent (parabolic) problems and to various nonlinear operators, playing a central role in existence, regularity, and qualitative analysis of solutions.