Sobolev Space - Motivation

Motivation

There are many criteria for smoothness of mathematical functions. The most basic criterion may be that of continuity. A stronger notion of smoothness is that of differentiability (because functions that are differentiable are also continuous) and a yet stronger notion of smoothness is that the derivative also be continuous (these functions are said to be of class C1 — see smooth function). Differentiable functions are important in many areas, and in particular for differential equations. On the other hand, quantities or properties of the underlying model of the differential equation are usually expressed in terms of integral norms, rather than the uniform norm. A typical example is measuring the energy of a temperature or velocity distribution by an L2-norm. It is therefore important to develop a tool for differentiating Lebesgue functions.

The integration by parts formula yields that for every u ∈ Ck(Ω), where k is a natural number and for all infinitely differentiable functions with compact support φ ∈ C_c∞(Ω),

,

where α a multi-index of order |α| = k and Ω is an open subset in ℝn. Here, the notation

is used.

The left-hand side of this equation still makes sense if we only assume u to be locally integrable. If there exists a locally integrable function v, such that

we call v the weak α-th partial derivative of u. If there exists a weak α-th partial derivative of u, then it is uniquely defined almost everywhere. On the other hand, if u ∈ Ck(Ω), then the classical and the weak derivative coincide. Thus, if v is a weak α-th partial derivative of u, we may denote it by Dαu := v.

The Sobolev spaces Wk,p(Ω) combine the concepts of weak differentiability and Lebesgue norms.