Sobolev space: I

by Yi Zhang

Sobolev space W_p^l is based on generalize derivatives and the corresponding norm induced. As a special case, space H^l has the meaning of finite “intensity” in general physical sense. For example, if u(x) is the flow velocity in spatial domain \Omega, then u\in H^1(\Omega) means the function and its first order derivatives are all in L^2(\Omega), i.e.

\int_\Omega{\nabla u \cdot \nabla u+u^2}d\Omega

Physically it means the total intensity of the sources within \Omega is finite, so is the total kinetic energy within the domain. Obviously this requirement is fundamental for any engineering analysis.

Meyers and Serrin proved that W_p^l\bigcap C^\infty is dense in W_p^l under the  Sobolev space norm.  This means that the function in Sobolev space can be approximated using smooth functions, and in many occasions the properties of Sobolev space is obtained by first demonstrate it in C^\infty.

The fundamental question of approximating the variational form of a PDE is twofold:

How much the smoothness/differentiability one can have from the weak solution?

How close the boundary condition can be approximated?

The first question is referred to as regularity problem. The properties of embedding, and traces of the Sobolev spaces are fundamental to answer questions above. It must be emphasized that the regularity of a weak solution Lu=f depends on the data f, the geometry of domain, and the boundary condition. All three factors must be examined in practical finite element convergence analysis.