Distributions and Sobolev Spaces

Overview¶

In classical analysis, differentiation requires smoothness. But PDE solutions are often non-smooth—think of shock waves, corners in domains, or fundamental solutions with point singularities. Distribution theory resolves this by shifting the burden of smoothness from the function to the test functions it acts on.

A distribution is a continuous linear functional on the space of smooth, compactly supported test functions $\mathcal{D}(\Omega)$. Every locally integrable function defines a distribution, but distributions also include singular objects like the Dirac delta:

This framework lets us:

• Differentiate any distribution, producing another distribution.

• Give meaning to “$-\Delta u = f$” even when $u$ is not twice differentiable.

• Build Sobolev spaces $W^{k,p}(\Omega)$ of functions with weak derivatives in $L^p$.

What You Will Learn¶

• Test functions, their topology, and the space $\mathcal{D}(\Omega)$.

• Distributions as continuous linear functionals on test functions.

• Operations on distributions: differentiation, multiplication, convolution.

• Mollifiers and approximation by smooth functions.