Weak Solution

A function (or equivalence class of functions) that satisfies a differential equation when integrated against a class of test functions, so that derivatives appear in the sense of distributions and boundary conditions are encoded via traces or the test space.

A generalized notion of solution in which the governing differential (or integral) equations are satisfied in an integrated or distributional sense against test functions, rather than pointwise; regularity requirements on the candidate function are relaxed so derivatives may exist only in a weak (distributional) sense.