Solvability compatibility condition

For an equation $Lu=f$ in a specified solution class, a compatibility condition is a condition on $f$ that is necessary for a solution to exist. In the linear setting, the complete abstract condition is $f\in\operatorname{im}L$ , the image of the operator with its chosen domain and boundary conditions.

Integral obstructions

If a linear functional $\ell$ vanishes on every $Lu$ in the chosen class, then $\ell(f)=0$ is necessary. For example, a compactly supported primitive $u$ on the real line satisfies $\int u'=0$ , so $u'=f$ requires $\int f=0$ . A list of necessary conditions is not automatically sufficient; sufficiency needs an inverse construction or a range theorem. Changing the support or boundary requirements can change the compatibility conditions.