In many areas of mathematics (PDE, Algebra, combinatorics, geometry) when we have difficulty in coming with a solution to a problem we consider various notions of "generalized solutions". (There are also other reasons to generalize the notion of a solution in various contexts.)
I would like to collect a list of "generalized solutions" concepts in various areas of mathematics, hoping that looking at these various concepts side-by side can be useful and interesting.
Let me demonstrate what I mean by an example from graph theory: A perfect matching in a graph is a set of disjoint edges such that every vertex is included in precisely one edge. A fractional perfect matching is an assignment of non negative weights to the edges so that for every vertex, the sum of weights is 1. In combinatorics, moving from a notion described by a 0-1 solution for a linear programming problem to the solution over the reals is called LP relaxation of a problem and it is quite important in various contexts.
(There are, of course, useful papers or other resources on generalized solutions in specific areas. It will be useful to have links to those but not as a substitute for actual answers with some details.)