Sobolev spaces: where equations find solutions that classical calculus cannot
Some important partial differential equations have no solutions among ordinary differentiable functions, yet do have them once derivatives are understood in a looser, weak sense. Sobolev spaces, named after the Russian mathematician Sergei Sobolev, are where such weak solutions live, with a norm that tracks both a function's size and its regularity.
Smoothness comes in grades: continuity, then differentiability, then a derivative that is itself continuous, and so on. For a long time these classical spaces seemed the natural home for solutions of differential equations, but in the twentieth century it became clear they were not quite right. Physical quantities in such models, like the energy of a temperature or velocity distribution, are usually expressed through integral norms, so a way to differentiate functions in Lebesgue spaces was needed.
The trick is integration by parts. For a differentiable function, integrating it against the derivative of any smooth test function with compact support equals, up to a sign, integrating the test function against its derivative. The first of those integrals still makes sense for functions that are merely locally integrable, so any function that makes the identity hold can be declared a weak derivative. It is unique almost everywhere, and when the classical derivative exists the two coincide.
A striking example is a tent-shaped function that climbs from minus 1 to 0 and descends to 1, but is assigned the value 10 at the single point zero. It is discontinuous at zero and not differentiable at minus 1, 0 or 1, yet it has a perfectly good weak derivative, equal to 1 on the left half and minus 1 on the right, and so belongs to a Sobolev space.
A Sobolev space bundles a function with its weak derivatives up to some order k and requires each to have a finite Lp norm, folding those norms into a single measure. Using weak derivatives makes the space complete, a Banach space. In one dimension it is enough to keep only the function and its highest derivative in the norm, and the spaces with p equal to 2 carry special importance.
Source: Sobolev space
