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.)
Best Answer
Partial Differential Equations (PDE) is a topic where generalizing the notion of solutions is a daily activity.
The most obvious generalization has been the notion of weak solutions, which means that a solution $u$ is not necessarily differentiable enough times for the derivatives involved in the equation to make sense; but an integration against test functions, followed by an integration by parts, cures the problem. The most known example is that of the Laplace equation $$\Delta u=f\qquad\hbox{over }\Omega,$$ where it is enough for $u$ to have locally integrable first-order derivatives, by rewriting the equation as a variational formulation (Dirichlet principle) $$\int_\Omega \nabla u\cdot\nabla vdx=-\int_\Omega fvdx$$ for every $v\in{\mathcal C}^1_c(\Omega)$ (subscript $c$ means compact support).
What is important in this process is to satisfy the rule
If $u$ has enough derivatives that the equation makes sense pointwise, then it is a weak solution if and only if it is a classical solution.
Let us mention in passing that in order to use the full strength of functional analysis and operator theory, this weak notion of solutions led to the birth of Sobolev spaces and Distribution theory (L. Schwartz).
This framework has been used for nonlinear equations and systems too, for instance for the Navier-Stokes, Euler, Schrödinger equations, ... An important question is whether this framework is accurate or not. By accurate, we mean that boundary and/or initial data yield a unique solution, which depends continuously on the data. This is the question of well-posedness. In many cases, functional analysis, sometimes associated to topological arguments, yield an existence theorem. A celebrated one is J. Leray's existence result to the Navier-Stokes equation of an incompressible fluid. However, uniqueness is often an other matter, a difficult one. For a $3$-dimensional fluid, the uniqueness to Navier-Stokes is a $1$M US Dollar open question.
Uniqueness is often (but not always) associated to regularity. In many situations, there are weak-strong uniqueness result, which state that if a classical, or a regular enough solution exists, then there does not exist any other weak solution (say in a class where we do have an existence result). It is an if-theorem, in the absence of an existence result of strong solutions. For elliptic and parabolic equations, the regularity theory is a topic of its own.
Whereas regularity is often expected in elliptic or parabolic equations and systems, it is not for hyperbolic ones, because we know that singularities do propagate, and that they can even be created in finite time thanks to nonlinear effects. Then the notion of weak solutions becomes meaningful, in that it translates in mathematical terms the physical notion of conserved quantities. It gives algebraic relation for the jump of the solution and its derivatives across discontinuities (Rankine-Hugoniot relations).
Finally, I like a lot the way the theory of nonlinear elliptic equations, and of Hamilton-Jacobi equations have develloped in the past decades. At the beginning, it was observed that the maximum principle, known for classical solutions, remains valid for weak ones. This suggested, when the nonlinearity is so strong that a variational formulation is not available, that the maximum principle itself be used to define a notion of viscosity solution. The idea is to test at $x_0$ the PDE with a test function $\phi$ being comparable to $u$ (either $\phi\le u$ or $\phi\ge u$ locally) and touching $u$ at $x_0$. This has been extremely powerfull.