There are numerous memes floating around with the words “Being weak is nothing to be ashamed of; staying weak is.” Or some variation. I thought about this meme in the context of weak derivatives.
The last couple posts have talked about distributions, also called generalized functions. The delta function, for example, is not actually a function but a generalized function, a linear functional on a space of test functions.
Distribution theory lets you take derivatives of functions that don’t have a derivative in the classical sense. View the function as a regular distribution, take its derivative as a distribution, and if this derivative is a regular distribution, that function is ca…
There are numerous memes floating around with the words “Being weak is nothing to be ashamed of; staying weak is.” Or some variation. I thought about this meme in the context of weak derivatives.
The last couple posts have talked about distributions, also called generalized functions. The delta function, for example, is not actually a function but a generalized function, a linear functional on a space of test functions.
Distribution theory lets you take derivatives of functions that don’t have a derivative in the classical sense. View the function as a regular distribution, take its derivative as a distribution, and if this derivative is a regular distribution, that function is called a weak derivative of the original function.
You can use distribution theory to complete a space of functions analogous to how the real numbers complete the rational numbers.
To show that an equation has a rational solution, you might first show that it has a real solution, then show that the real solution is in fact a rational. To state the strategy more abstractly, to find a solution in a small space, you first look for solutions in a larger space where solutions are easier to find. Then you see whether the solution you found lies in the smaller space.
This is the modern strategy for studying differential equations. You first show that a differential equation has a solution in a weak sense, then if possible prove a regularity result that shows the solution is a classical solution. There’s no shame in finding a weak solution. But from a classical perspective, there’s shame in stopping there.